X-Ray Charge Densities and Chemical Bonding

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INTERNATIONAL UNION OF CRYSTALLOGRAPHY TEXTS ON C R Y S T A L L O G R A P H Y

l U C r BOOK S E R I E S C O M M I T T E E E. N. Baker, Palmerston A. A. Chernov, Moscow P. Coppens (Chairman), Buffalo G. Desiraju, Hyderabad J. Drenth, Groningen J. Helliwell, Manchester S. Larsen, Copenhagen

I N T E R N A T I O N A L UNION OF CRYSTALLOGRAPHY BOOK S E R I E S

This volume forms part of a series of books sponsored by the International Union of Crystallography (lUCr) and published by Oxford University Press. There are three lUCr series: lUCr Monographs on Crystallography, which are in-depth expositions of specialized topics in crystallography; lUCr Texts on Crystallography, which are more general works intended to make crystallographic insights available to a wider audience than the community of crystallographers themselves; and lUCr Crystallographic Symposia, which are essentially the edited proceedings of workshops or similar meetings supported by the lUCr. lUCr Monographs on Crystallography 1 Accurate molecular structures: Their determination and importance A. Domenicano and I. Hargittai, editors 2 P. P. Ewald and his dynamical theory of X-ray diffraction D. W. J. Cruickshank, H. J. Juretscke, and N. Kato, editors 3 Electron diffraction techniques, Volume 1 J. M. Cowley, editor 4 Electron diffraction techniques, Volume 2 }. M. Cowley, editor 5 The Rietveld method R. A. Young, editor lUCr Texts on Crystallography 1 The solid state: From superconductors to superalloys A. Guinier and R. Jullien, translated by W. J. Duffin 2 Fundamentals of crystallography C. Giacovazzo, editor 3 The basics of crystallography and diffraction C. Hammond 4 X-ray charge densities and chemical bonding P. Coppens lUCr Crystallographic Symposia 1 Patterson and Pattersons: Fifty years of the Patterson function J. P. Glusker, B. K. Patterson, and M. Rossi, editors 2 Molecular structure: Chemical reactivity and biological activity J. J. Stezowski, J. Huang, and M. Shao, editors 3 Crystallographic computing 4: Techniques and new technologies N. W. Isaacs and M. R. Taylor, editors 4 Organic crystal chemistry i. Garbarczyk and D. W. Jones, editors 5 Crystallographic computing 5: From chemistry to biology D. Moras, A. D. Podjarny, and J. C. Thierry, editors 6 Crystallographic computing 6: A window on modern crystallography H. D. Flack, L. Parkanyi, and K. Simon, editors 7 Correlations, transformations, and interactions in organic crystal chemistry D. W. Jones and A. Katrusiak, editors

X-RAY CHARGE DENSITIES AND CHEMICAL BONDING

Philip Coppens Chemistry Department State University of New York at Buffalo

International Union of Crystallography Oxford University Press 1997

Oxford University Press Oxford New York Athens Auckland Bangkok Bogota Bombay Buenos Aires Calcutta Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin Ibadan

Copyright © 1997 by Oxford University Press, Inc. Published by Oxford University Press, Inc., 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Coppens, Philip. X-ray charge densities and chemical bonding / Philip Coppens. p. cm. — (International Union of Crystallography texts on crystallography ; 4 Includes bibliographic references and index. ISBN 0-19-509823-4 1. X-ray crystallography. 2. Chemical bonds. 3. Electron distribution. I. Title. II. Series QD945.C597 1997 548'.3— dc21 96-48736

3 5 7 9 8 6 4 2 Printed in the United States of America on acid-free paper

Preface It seems to me that the experimental study of scattered radiation, in particular from light atoms, should get more attention, since along this way it should be possible to determine the arrangement of the electrons in the atoms P. Debye(1915) An accurate set of nuclear coordinates and a detailed map of the electron density can be obtained, by X-ray diffraction, only jointly and simultaneously, never separately or independently F. L. Hirshfeld (1992)

The ability to measure the experimental charge distribution in crystals from the intensities of the scattered X-rays was realized almost immediately after the discovery of X-ray diffraction. Notwithstanding this early recognition, the technical developments of the 1960s and beyond, which occurred in diffractometry, automation of data collection, low-temperature techniques, and computers, were needed to achieve a breakthrough in the method. The accurate crystallographic methods developed during the past decades led not only to a much better precision in atomic coordinates, but also to crucial information on the charge distribution in crystals. This experimentally obtained distribution can be compared directly with theoretical results, and can be used to derive other physical properties, such as electrostatic moments, the electrostatic potential, and lattice energies, which are accessible by spectroscopic and thermodynamic measurements. This broad interface with other physical sciences is one of the most appealing aspects of the field. The aim of this volume is to provide the background necessary for interpretation of the results of accurate crystallographic methods, and to present the concepts to a wider community of nonspecialized scientists. Though a number of excellent conference proceedings exist, there is no single text summarizing the subject. This text is not meant as a comprehensive review of the existing literature; experimental results are presented as an illustration of principles and methods. While this leaves out a great many valuable studies available in the literature, the restraint is unavoidable if the text is to be of manageable size. Experimental methods have not been covered, as they are still developing rapidly. New developments, in particular the advent of third-generation synchrotron sources combined with parallel data collection methods and new interpretative software packages, are bound to further enhance the scope of the field. It is hoped that this text will serve to stimulate its continuing development. I am grateful to the many colleagues who have commented on the contents

vi

Preface

of this volume, and especially to Dr. Zhengwei Su who played a key role in the development of the formalisms described in chapters 7-9 and appendixes D, E, G, and H, and who has pointed out many errors and omissions in earlier versions of the text. Others, including D. Feil, T. R. Furlani, C. Lecomte, V. Petricek, and S. Price provided additional valuable suggestions and corrections. I would like to thank S. Priore-Fensore for her outstanding dedication, which made the completion of the manuscript possible, and I. Novozhilova for assistance in the final stages of this project. Philip Coppens Buffalo, NY November 1996

Contents

Chapter 1 Scattering of X-rays and Neutrons

3

1.1 Outline of this chapter 3 1.2 Introduction to the theory of X-ray scattering 3 1.2.1 Classical treatment of X-ray scattering 3 1.2.2 Quantum-mechanical treatment: the first Born approximation 5 1.2.3 Scattering by a periodic crystal 7 1.2.4 The structure factor formalism in terms of atomic densities 9 1.3 Resonance scattering of X-rays 11 1.3.1 Classical treatment 11 1.3.2 Quantum-mechanical treatment: the second Born approximation 13 1.3.3 The power series expansion of the scattering operator 15 1.3.4 The optical theorem and the relation between f" and f' 16 1.4 Neutron scattering 18 1.4.1 Properties of neutrons 18 1.4.2 The neutron scattering length 19 Chapter 2 The Effect of Thermal Vibrations on the Intensities of the Diffracted Beams 22 2.1 The normal modes of a crystal 23 2.1.1 Phonons, internal and external modes 23 2.1.2 The frequency of the normal modes 23 2.2 The effect of thermal vibrations on the Bragg intensities 27 2.2.1 The Born-Oppenheimer approximation 27 2.2.2 The harmonic temperature factor 28

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2.2.3 Beyond the harmonic approximation 31 2.2.4 Comparison of the anharmonic formalisms 36 2.2.5 Quantum-statistical treatments 37 2.3 The relation between the atomic temperature factors and lattice dynamics 40 2.3.1 General expression 40 2.3.2 The Debye approximation 41 2.3.3 The rigid-body model for molecular crystals 42 2.3.4 The rigid-bond test 48 Chapter 3 Chemical Bonding and the X-ray Scattering Formalism

49

3.1 The breakdown of the independent-atom model 49 3.1.1 Qualitative considerations 49 3.1.2 The electron density and the LCAO formalism 51 3.2 Improved scattering models 55 3.2.1 The spherical atom kappa formalism 55 3.2.2 Modified spherical scattering factor for the hydrogen atom 56 3.2.3 Examples of results obtained with the /c-formalism 57 3.2.4 The multipole description of the charge density of aspherical atoms 59 3.2.5 Aspherical atom scattering factors 67 3.2.6 The aspherical density functions of the Hirshfeld formalism 70 Chapter 4 Least-Squares Methods and Their Use in Charge Density Analysis 72 4.1 Least-squares equations 72 4.1.1 Background 72 4.1.2 General formalisms for linear least-squares 72 4.1.3 Explicit expressions for structure factor least-squares 74 4.1.4 Variances and covariances of the least-squares parameter estimates 76 4.1.5 Uncorrelated linear combinations of variables 79 4.2 The least-squares parameters in charge density analysis 79 4.2.1 The parameters in a charge density refinement 79 4.2.2 Parameter restrictions imposed by site and local symmetry and chemical equivalence 80 4.2.3 The scale factor 81 4.3 Physical constraints of the electron density 83 4.3.1 The electroneutrality constraint 83 4.3.2 The Hellmann-Feynman constraint 85 4.4 Joint refinement of X-ray and neutron data 86 4.4.1 The use of complementary information 86 4.4.2 Differences in temperature parameters 86 4.4.3 Relative weighting of the X-ray and neutron data 87 4.4.4 Estimate of the goodness of fit 88

Contents

ix

Chapter 5 Fourier Methods and Maximum Entropy Enhancement 90 5.1 General expressions 90 5.1.1 The total density 90 5.1.2 The residual density 92 5.1.3 Least-squares minimization and the residual density 93 5.2 Deformation densities 94 5.2.1 Definition of the deformation density 94 5.2.2 The choice of a model for the reference state 95 5.2.3 The choice of structural parameters in the calculation of the deformation density 101 5.2.4 Combining Fourier and least-squares methods: the model deformation density 105 5.2.5 Phase contributions to the deformation density 109 5.2.6 The variances and covariances of the experimental density 111 5.3 Maximum entropy enhancement of electron densities 115 5.3.1 The mathematics of entropy maximization 115 5.3.2 Application of MEM to single-crystal data on silicon 117 5.3.3 The two-channel maximum entropy method 118 Chapter 6 Space Partitioning and Topological Analysis of the Total Charge Density 121 6.1 Methods of space partitioning 122 6.1.1 The stockholder partitioning concept 122 6.1.2 Space partitioning based on the atom-centered multipole expansion 124 6.1.3 Atomic fragments defined by discrete boundaries 124 6.2 Space partitioning based on the topology of the total electron density 130 6.2.1 The definition of critical points 130 6.2.2 The surface of zero flux 132 6.3 Chemical bonding and the topology of the total electron density distribution 134 6.3.1 The Laplacian of the electron density 134 6.3.2 Classification of bonds based on the topology of the electron density 136 6.3.3 Topological analysis of experimental densities 139 Chapter 7 The Electrostatic Moments of a Charge Distribution 142 7.1 Moments of a charge distribution 143 7.1.1 Definitions 143 7.1.2 The origin dependence of the electrostatic moments 145 7.2 Electrostatic moments from diffraction data 147 7.2.1 Atomic electrostatic moments in terms of the parameters of the multipole formalism 147 7.2.2 Molecular moments as a sum over the pseudoatom moments 149

x

Contents

7.2.3 The electrostatic moments of the deformation density 149 7.2.4 Electrostatic moments of a subvolume of space by Fourier summation 151 7.2.5 Error analysis of diffraction moments 154 7.3 Discussion of experimental results 155 7.3.1 A compilation of X-ray electrostatic moments 155 7.3.2 Dipole moments 160 7.3.3 Second moments and quadrupole moments 163 Chapter 8 X-ray Diffraction and the Electrostatic Potential

165

8.1 Definitions and units 166 8.1.1 Definition of the electrostatic potential 166 8.1.2 The electric field 167 8.1.3 The electric field gradient 167 8.1.4 Units 168 8.2 Evaluation of the electrostatic potential and its derivatives in reciprocal space 168 8.2.1 The electrostatic potential by Fourier summation of the structure factors 168 8.2.2 The O(0) term in the potential summation 170 8.2.3 The electric field and the electric field gradient by direct Fourier summation of the structure factors 172 8.2.4 Application of the AF summation to the electrostatic potential in Lalanine 174 8.3 Evaluation of the electrostatic functions in direct space 174 8.3.1 Basic expressions 174 8.3.2 Analytical evaluation of the electrostatic properties due to the spherical components of the charge density 177 8.3.3 Electrostatic properties and the multipole expansion 177 8.4 Comparison of diffraction results with theory and other experimental values 181 8.4.1 The comparison of experimental and theoretical potentials 181 8.4.2 The relation between the electrostatic potential at the nuclear position, the Is binding energy and the net atomic charge 183 8.4.3 The electric field gradient at the nuclear position 184 8.5 The electrostatic potential outside a charge distribution 186 8.5.1 The electrostatic potential outside a charge distribution in terms of the multipole moments 186 8.5.2 Net atomic charges reproducing the electrostatic potential 186 Chapter 9 The Electron Density and the Lattice Energy of Crystals 192 9.1 The total energy of a system 192 9.2 Density functional expressions for the energy

192

Contents

xi

9.2.1 The Hohenberg-Kohn theorem 192 9.2.2 Density functional expressions 193 9.2.3 The total energy as a function of tne electrostatic potential at the nuclear position 194 9.3 The cohesive energy of ionic crystals 195 9.3.1 The point-charge model 195 9.3.2 Fourier series for the total electrostatic energy 196 9.3.3 The accelerated convergence method and the electrostatic potential in a point-charge crystal 196 9.3.4 The lattice energy of a crystal consisting of spherical ions 198 9.3.5 Application to the lattice energy of alkali halides 200 9.4 The cohesive energy of molecular crystals 203 9.4.1 Molecular interactions 203 9.4.2 Interactions between molecules in crystals 206 9.4.3 Expressions for the evaluation of the electrostatic contribution to the lattice energy of molecular crystals 206 9.4.4 Calculated lattice energies of molecular crystals 208 Chapter 10 Charge Density Studies of Transition Metal Compounds

211

10.1 The study of transition metal complexes 211 10.2 On the electronic structure of transition metal atoms 212 10.2.1 Crystal field splitting of d orbitals 212 10.2.2 Effect of covalency on the orbital populations 214 10.2.3 The relation between the occupancies of transition-metal valence orbitals and the multipole population parameters 216 10.3 The electric field gradient at the nucleus of a transition metal atom 220 10.3.1 Electric field gradient expressions for transition metal elements 220 10.3.2 Comparison with results from Mossbauer spectroscopy 221 10.4 Electron density studies of octahedral and distorted octahedral complexes 227 10.4.1 Complexes with CO, CN, and NH3 ligands 227 10.4.2 First-row transition metal sulfides 228 10.5 Electron density studies of iron(II) porphyrins 230 10.5.1 The electronic structure of iron(II) tetraphenyl porphyrins 230 10.5.2 Experimental results for iron porphyrins and comparison with theory 230 10.6 The electron density in metal-metal bonds of transition metal complexes 238 10.6.1 Metal-metal bonding 238 10.6.2 Bonding between chromium atoms 238 10.6.3 Mn—Mn and Fe—Fe bonding 240 10.7 Anharmonic thermal motion and the bonding anisotropy of transition metal complexes 241

xii

Contents

10.7.1 Atomic asphericity and anharmonic thermal motion 241 10.7.2 A model study on theoretical structure factors of Fe(H2O)6 242 10.7.3 An experimental example: anharmonic refinement of bis(pyridine) (meso-tetraphenylporphinato) iron(II) 244 Chapter 11 The Charge Density in Extended Solids

247

11.1 Covalently bonded extended solids 248 11.1.1 Silicon 248 11.1.2 Charge density studies of silicates 253 11.2 Metallic solids 257 11.2.1 Specific aspects of metals 257 11.2.2 The charge density in beryllium metal 258 11.2.3 The charge density in vanadium and chromium 261 11.2.4 The charge density in copper 264 11.2.5 Intermetallic compounds and the critical-voltage electron diffraction method 265 11.3 Ionic solids 268 11.3.1 Do completely ionic solids exist? 268 Chapter 12 Electron Density Studies of Molecular Crystals

271

12.1 Why molecular crystals? 271 12.1.1 The importance of molecular crystals 271 12.1.2 The suitability factor 272 12.1.3 The oxalic acid project 273 12.2 Transferability of charge density parameters among related atoms 273 12.3 Selected studies of molecular crystals 277 12.3.1 Bent bonds 277 12.3.2 Molecular crystals with nonlinear optical properties 282 12.3.3 The effect of intermodular interactions on the charge density 283 12.4 Concluding remarks 287 Appendix Appendix Appendix Appendix Appendix Appendix

A B C D E F

Tensor Notation 288 Symmetry and Symmetry Restrictions 290 The 50% Probability Ellipsoid 295 Spherical Harmonic Functions 297 Products of Spherical Harmonic Functions 307 Energy-Optimized Single- Slater Values for Subshells of Isolated Atoms 311 Appendix G Fourier-Bessel Transforms 313 Appendix H Evaluation of the Integrals AN h hyk(Z, R) Occurring in the Expression for the Peripheral Contribution to the Electrostatic Properties 315

Contents

xiii

Appendix I The Matrix M-1 Relating rf-Orbital Occupancies Pij to Multipole Populations Plmp 317 Appendix J The Interaction Between Two Nonoverlapping Charge Distributions 318 Appendix K Conversion Factors 320 Appendix L Selected Exercises 320 References Index

347

328

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X-RAY CHARGE DENSITIES AND CHEMICAL BONDING

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1 Scattering of X-rays and Neutrons

1.1 Outline of this Chapter

This chapter starts with a discussion of the classical treatment of X-ray scattering, followed by a brief overview of the quantum-mechanical theory in the first Born approximation. The scattering of a periodic arrangement is derived by considering the crystal as a convolution of the unit cell contents and a periodic lattice. The atomic description of the charge density, which is the basis for structure analysis, is introduced. The origin of resonance anomalous scattering is discussed. While its effect must be accounted for before charge densities can be derived from the X-ray scattering amplitudes, resonance scattering itself can give invaluable information on the electronic states of the resonating atoms. The final section of this chapter deals with the scattering of neutrons by atomic nuclei. Nuclear neutron scattering is independent of the distribution of the electrons, and can provide atomic positions and thermal amplitudes unbiased by the bonding effects which are the subject of this book.

1.2 Introduction to the Theory of X-ray Scattering 1.2.1 Classical Treatment of X-ray Scattering

In the classical theory of scattering (Cohen-Tannoudji et al. 1977, James 1982), atoms are considered to scatter as dipole oscillators with definite natural frequencies. They undergo harmonic vibrations in the electromagnetic field, and emit radiation as a result of the oscillations. The equation of motion for a single harmonic oscillators of mass m, and force 3

4

X-ray Charge Densities and Chemical B o n d i n g

constant kf (kf = mcos2), is (following Newton) given by where x is the second time derivative of the displacement x. When the oscillator is a particle with charge —e, exposed to an oscillating electric field one obtains

When the oscillation is damped, with damping factor k, and damping /ex proportional to the speed of motion x, one obtains

The corresponding time-dependent value of the damped oscillating dipole equals

as can be verified by direct substitution of Eq. (1.4) into Eq. (1.3). The oscillating dipole is a source of electromagnetic radiation of the same frequency, polarized in the direction of the oscillations. At large distances, the wave is spherical. According to the electromagnetic theory, the resulting electric vector at a point in the equatorial plane of the dipole is times the moment of the dipole at time t — |r|/c. The amplitude of the spherically scattered wave at unit distance in the equatorial plane is therefore

The scattering of the free electron for E0 = 1 is obtained from this expression by setting both the force constant (a,, and the damping factor k equal to zero, which gives The quantity e2/mc2 is the scattering amplitude of the classical electron, denoted by the symbol re, and generally used as the unit of electron scattering. Its numerical value equals 2.818-10" 13 cm = 2.818 fermi. 1 For an assembly of several free point electrons, interference occurs between radiation scattered by different centers. If the incident and diffracted beams are defined by two unit vectors, s0 and s, respectively (Fig. 1.1), the phase difference of the radiation scattered by two points, separated by the vector r, equals 27cS-r, where S is the scattering vector, equal to (s — s0)/A. Vector S bisects s and s0, and has the length 2 sin Q/L In the physics literature, an alternative notion is commonly used. The incident and diffracted beams are defined by the vectors k

1 If physical constants in SI units are used in the evaluation of e2/mc2, the result has to be divided by the factor 4n times the permittivity of free space 4m:0 = 1.1126265-10" 10 C 2 N~ ' m~ 2

Scattering of X-rays and Neutrons

5

FIG. 1.1 Geometry of scattering

and k0, of length 2n//., such that the scattering vector K = k - k0 has a magnitude |K | + 47rsin0/;.. For the scattering by two point electrons at distance r, using the rules for addition of coherent waves, we get per unit incident amplitude £0

in which the negative sign of Eq. (1.5b) has been factored out. For a continuous electron distribution p(r), the summation over waves of different phase must be replaced by an integration leading to a diffraction amplitude:

Thus, for co » a>s and w2 » ikta, the amplitude of scattering is the Fourier transform of the electron density. This important result is confirmed by the first-order quantum-mechanical treatment discussed in the next section. When w s cus, both the amplitude and the phase of the scattered radiation depend on the frequency. 1.2.2 Quantum-Mechanical Treatment: The First Born Approximation The treatment of the interaction of a quantum-mechanical system with radiation is described in detail in the literature (Feil 1975; Cohen-Tannoudji et al. 1977; Blume 1985, 1994). Only an outline will be given here. The Hamiltonian H of a system in a radiation field, in the absence of interaction with the field and a perturbation Hamiltonian H' describing the interaction with the field, is given by The interaction of a system with the electromagnetic radiation leads to a :ime-dependence of the wave function ij/, which follows from the time-dependent Schrodinger equation

6

X-ray Charge Densities and Chemical Bonding

FIG. 1.2 Feynman diagrams describing (a) elastic scattering, (b) inelastic scattering. Source: Feil (1975).

The time-dependent wave function is described by in which the x values are the coordinates of the scattering particles, and the transition is from an initial state ij/m to a final state \j/n. The wave function i// describes both the scattering assembly M and the radiation field; in shorthand notation, for the initial-state wave function and an incident photon with propagation vector In the first Born approximation, the interaction between the photons and the scattering system is weak and no excited states are involved in the elastic scattering process. Furthermore, there is no rescattering of the scattered wave, that is, the single-scattering approximation is valid. In the Feynman diagrams (Fig. 1.2), there is only one point of interaction for first-Born-approximation processes. The interaction Hamiltonian contains the operator A, corresponding to the vector potential A of the electromagnetic field.2 Excluding magnetic scattering, the interaction Hamiltonian is given by

in which q is the charge of the scattering particle, and m is its mass. Note that scattering by the heavier nuclei is orders of magnitude smaller than the scattering by the electrons. Vector p is the momentum operator equal to The second term in Eq. (1.11) is the origin of the scattering in the first Born approximation. It leads to an amplitude for the scattering of photons with propagation vector k0 into photons with vector k equal to with K = k — k0. In this expression, is the wave function describing the assembly only, and C is a proportionality constant which includes the polarization factor. 3 The positive sign of the exponent in Eq. (1.12) is the convention followed in crystallography (James 1982) and adopted in this text. To avoid confusion, it 2

The electric field E and the magnetic field B are related to the vector potential by B = V x A. See, for example, Jackson (1977). 3 Polarization is explicitly included in the treatment in section 1.3.2.

and


7

should be pointed out that in the physics literature the plane-wave expression exp ( —;iK-r) is used. For independently scattering systems, without specific phase relation, we get for the total intensity in the scattered beam

The integration is over the coordinates of all the electrons, and the wave function ty describes all particles of the scattering system. For elastic scattering, illustrated by Fig. 1.2(a), the initial and final states are the same, that is, n = m. For a system of electrons, represented by a many-electron wave function, we obtain in the approximation that the N electrons are scattering indeoendentlv:

As the electrons are indistinguishable in the antisymmetrized wave function, the one-electron scattering can be obtained by integration over all coordinates but those of the jth electron. Summation over all equivalent electrons then leads to

where p(r0 is the electron distribution Or, for the scattering amplitude A(K),

where F is the Fourier transform operator. This result is equivalent to the classical expression (1.7). It is sometimes referred to as the form-factor approximation, and is based on the assumption that the particles scatter independently of each other. This assumption neglects binding effects among the electrons and between electrons and nucleons, which become important as very low and very high energies (Kissel et al. 1995). 1.2.3 Scattering by a Periodic Crystal According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,

where n, m, and p are integers, and 6 is the Dirac delta function. Equation (1.16) states that the scattering of the crystal is the Fourier transform of pcrystal(r). According to the Fourier convolution theorem, the Fourier transform of a convolution of two functions is the product of the Fourier transform of the individual functions, or

8


The Fourier transform of the direct space 6 function is a 6 function in reciprocal space, representing the reciprocal lattice. We thus obtain

It follows from Eq. (1.19) that a crystal scatters with an amplitude proportional to F{p U nitceii( r )} m directions defined by the scattering vectors where the reciprocal axes a* j = 1

3

are defined by

Thus, the scattering of a periodic lattice occurs in discrete directions. The larger the translation vectors defining the lattice, the smaller a* i = 1 3, and the more closely spaced the diffracted beams. This inverse relationship is a characteristic property of the Fourier transform operation. The scattering vectors terminate at the points of the reciprocal lattice with basis vectors af i= l 3, defined by Eq. (1.21). The Fourier transform of the unit cell density F{pumtceii(r)} is referred to as the structure factor F:

Expression (1.19) is valid for a crystal with a very large number of unit cells, for which particle-size broadening is negligible, as indicated by the infinitely sharp , is referred to as the Fourier-Bessel transform of the atomic density. For an atomic density expressed as a sum over exponential functions, Eq. (1.28) can be expressed in closed form as discussed in chapter 3. The appropriate expressions are listed in appendix G. Scattering factors for the isoelectronic F~ and Na"1" ions are shown in Fig. 1.5. The inverse relationship between direct space and reciprocal space is evident, as the more compact Na+ ion has the more "diffuse" scattering factor. For the same reason, the scattering of the atomic valence electrons is concentrated in the low-order region of reciprocal space, while the core electron scattering persists to high values of sin 9/L This is the basis for the high-order refinement method, in which only high-order data are used to reduce the bias in the structural parameters due to deviations from the IAM. The high-order refinement method is further discussed in chapter 3. 1.3 Resonance Scattering of X-rays

1.3.1 Classical Treatment The classical scattering amplitude of an electron was derived in section 1.2.1 as

12

X - r a y Charge Densities and Chemical Bonding

The amplitude A can be separated into real and imaginary parts by multiplication of both the numerator and the denominator by (a2 — to2 — ika>. For unit value of E0, in units of —e/mc2, the result is

The anomalous contribution to the real part of the scattering amplitude can be separated by subtraction of the classical Thompson scattering, — e2/mc2, from the first term of Eq. (1.29) to give

According to Eqs. (1.29) and (1.30), the real part of the resonance scattering amplitude of an oscillator is negative below the absorption edge, where u> < a> and positive above the edge, while the imaginary part of the amplitude is negative everywhere (i.e., has a sign opposite to that of the classical electron scattering). This conclusion is confirmed by more advanced theory and by experiment, except in the vicinity of the absorption edges, where the behavior is more complicated than can be accounted for by the simple classical theory. Including resonance effects, the atomic scattering factor for a many-electron atom is written as

where the last two terms describe the real and imaginary parts of the resonance contribution, respectively. In the classical theory, the electron scattering for each electron shell is multiplied by the oscillator strength g(s) of the shell. Separation of the real and imaginary parts in units of scattering of the classical electron, leads to

and

These equations neglect the angle dependence of the atomic scattering and thus assume that the atomic dimensions are small relative to the wavelength of the photons. This is a good approximation for the dimensions of the atomic shells and the resonance wavelengths of the corresponding absorption edges. For values of that is, for negligible effect of the resonance, the real contribution Thus, the real contribution becomes equal to the oscillator strength. As discussed by James (1982), there is a close correspondence between the classical oscillator strengths and the matrix element of quantum-mechanical scattering theory.


13

1.3.2 Quantum-Mechanical Treatment: The Second Born Approximation In the vicinity of the atomic absorption edges, the participation of free and bound excited states in the scattering process can no longer be ignored. The first term in the interaction Hamiltonian of Eq. (1.11) leads, in second-order perturbation theory, to a resonance scattering contribution (in units of classical electron scattering) equal to (Gerward et al. 1979, Blume 1994)4

where p is the electron momentum operator, and m is the electron mass, the wave function of the initial (and final) state including the photon, and e0 and e, are the polarization vectors of the incident and reflected beams, respectively. The summation is over all unoccupied orbitals, including the free states in the continuum. For the latter, the summation must be replaced by an integration. The two terms in Eq. (1.34a) include processes in which the initial photon k0 has been annihilated first, and those in which the final photon k has first been created. In the quantum-mechanical description of the first type of process, the photon k0 is absorbed and then, in a very small time interval, the photon k is emitted through a stimulated emission process. This process and three-beam multiple scattering are illustrated by the Feynman diagrams in Fig. 1.6. The first term in Eq. (1.34a) represents the resonance scattering. It becomes large when In comparison, the second term is small and is usually neglected. The imaginary term in the denominator contains the inverse lifetime (related to linewidth) of the intermediate state For a system of independently scattering electrons, the resonance scattering amplitude, ignoring the second term, becomes

(1.34b) where the sum in the numerator is over all the electrons. The terms in the summation for which __ correspond to an absorption of radiation and give rise to the imaginary part of the dispersion /". 4

To be consistent with the physics literature, in this section the incident photon wave function is defined as exp(ik 0 -r), rather than as exp ( — ;k 0 -r).

14


FIG. 1.6 Feynman diagrams for (a) multiple scattering, and (b) resonance scattering. Source: Feil (1975).

The separation between the real and imaginary components is achieved by use of the expression

where P indicates that the term is excluded when x = 0. The real part is then

where P represents the principal part of the summation in the sense that all terms with zero denominator are left out, while the imaginary part /" is given by

Using

for the components of the momentum operator p, and

the term denned by the numerator in Eq. (1.36) and the expression in square brackets in Eq. (1.37) can be reduced to

provided the operators between the vertical lines in Eqs. (1.36) and (1.37) are Hermitian (Feil 1992). The factor Cj -e0 is the polarization vector routinely applied in the reduction of X-ray intensities to structure factor amplitudes.


15

As is evident from Eqs. (1.36) and (1.37), and from the classical treatment as well, the effect of resonance on the intensity of X-ray scattering is pronounced when that is, in the vicinity of the absorption edges. Even for data collected at other wavelengths, it is necessary to correct the structure factors for anomalous scattering before the electron density can be calculated by the Fourier inversions of Eqs. (1.22) and (1.26), as further discussed in chapter 5. The anomalous scattering factors needed for this purpose are available in the literature (International Tables for X-ray Crystallography 1974, Kissel and Pratt 1990). The position of the absorption edge, and its fine structure, give information on the ionization energy of the resonating electrons, and on the nature of spectroscopic transitions from the resonating shell to unoccupied bound states of the scattering entity (Pickering et al. 1993, Sorensen et al. 1994). The ionization energy / is, in turn, related to the binding energy e of the electrons and to the electrostatic potential $ at the atomic nucleus, which is one of the physical quantities that can be derived from the charge distribution (see chapter 8 for a discussion of the relation between /, and ). 1.3.3 The Power Series Expansion of the Scattering Operator Scattering operators of the form exp (ik-r), as occur in Eqs. (1.34)-(1.40), may be developed in a power series:

The approximation in which only the leading term in the expansion is retained is referred to as the dipolar approximation. The dipolar approximation will be more closely obeyed for small values of that is, for longer wavelengths and, in particular, for visible light. The higher-order terms are also smaller when the scattering object is compact relative to the wavelength used, that is, if either the initial state or the final state has a compact core-type wave function. In the dipolar approximation, the matrix elements in Eq. (1.40) become equal to for a polarization direction defined by the unit vector e. The anomalous scattering effects are then independent of the magnitude of K = k — k0. Factoring out the polarization effects implicit in e,-e 0 , the angular dependence of the scattering factor is described by the products 7)rt. In other words, the scattering amplitude is a second-order tensor property when either the ground state or the excited stated wave function is not spherically symmetric. Retention of additional terms of the expansion Eq. (1.41) in Eq. (1.40) gives rise to a more complex angular dependence. Templeton has formulated the total resonance scattering in terms of a tensor equation, which, including terms up to fourth order, is given by (Templeton 1994)

16


where the primed quantities refer to the scattered beam, and summation over repeated indices is assumed. The imaginary Q24 term is a mixed dipolarquadrupolar term, which is absent for atoms at centrosymmetric sites. A second fourth-rank contribution is due to the dipolar-octapolar term Q 2 8 - Experimental evidence for the Q 24 term has been obtained from intensity measurements on potassium chromate (Templeton and Templeton 1993). The existence of the Q44 44 term was demonstrated in a study of the forbidden (111) reflection in hematite (x-Fe2O3) (Finkelstein et al. 1992). 1.3.4 The Optical Theorem and the Relation Between f"and f' The scattering of the X-ray beam in the forward direction adds out-of-phase components to the propagating beam, as the classical electron scattering has a negative sign. This implies that the X-ray refractive index n differs from unity. The reduction in n leads to total reflection at very small angles, which is applied in the design of X-ray mirrors. Since the scattering amplitude has both real and imaginary components, we may write Here, x, the real component of 6, is related to the atomic scattering factors / by the expression where the sum is over all atom types, each with number density Ni and /• and /{ are expressed in units of the classical electron scattering. An equivalent formula is obtained by substitution of where ca is the angular frequency

The complex term , in Eq. (1.41) depends on /" in a manner analogous to that expressed by Eq. (1.44): Since and /" represent an absorption of the propagating beam, they are related to the linear absorption coefficient This relation is called the optical theorem At energies just above an absorption edge, the contribution of the strongly absorbing atom is dominant. The absorption of other atoms is often small and in first approximation is independent of the energy. Expression (1.47) can be used to obtain for the dominantly resonating atom, over a large energy range, from the experimental absorption curve. Substitution of the classical scattering amplitude e2/mc2, equal to 0.2818 • 10"14 m, in Eq. (1.47) gives a numerical relation between and f"(/.):


in which is in m" 1 , common units:

17

in m, and N, the number density, in m ~ 3 . Or, in more

The expression for the atomic cross section follows from The mass absorption coefficient of the element is given by where p is the density, and A and M are Avogadro's number and the atomic weight, respectively. This gives, analogously to Eq. (1.49),

The two anomalous components of the scattering factor, f" and f', are interrelated through the Kramers-Kronig transforms, which have the form

or, equivalently,

and for the inverse transformation

Since both expressions have a singularity at special mathematical techniques are used for precise evaluation. A symbol P is often inserted in front of the integral to indicate that it must be evaluated as the Cauchy principal value, excluding the infinite contribution at The proper mathematical procedure to integrate across the singularity has been discussed by Hoyt et al. (1984). The integration in the Kramers-Kronig expression is over an infinite range. Theoretical values of f" in regions remote from the absorption edge are therefore required for the application of the transform. They can be calculated as described by Cromer and Liberman (1970). The fitting of the absorption curve to the calculated values for the dominant scatterer provides a scale for f", and eliminates the slowly varying contributions by other atoms to Eq. (1.47) (see, e.g., Hendrickson et al. 1988). As a result of errors in the extrapolation procedure and the absorption curves, an uncertainty of 0.2-0.3 electrons in the height of the whole curve appears to be present. This should be compared with the total changes in /', which may have magnitudes of 6-10 electrons or larger. A typical relation between f' and f" near an absorption edge is illustrated in Fig. 1.7. It shows that /' has a minimum when f" has increased by about half its total variation, that is, halfway up the slope of the edge.

18


FIG. 1.7 Anomalous scattering terms f' and f" for gadolinium near the L3 edge. Source: Giacovazzo (1992). 1.4 Neutron Scattering 1.4.1 Properties of Neutrons The wavelength of the neutron is related to its mass m ( = 1.675-10" 27 kg) and velocity v by the de Broglie relation

The velocity distribution in a neutron gas at equilibrium is subject to the laws of the kinetic theory of gases. The neutron velocities at equilibrium obey the Maxwell distribution

where kB is Boltzmann's constant. From Eq. (1.54), the rms value of the velocity of the neutrons is equal to

which gives, for the average kinetic energy, Substituting Eq. (1.56) into Eq. (1.53) gives, for the average neutron wavelength,

As neutrons from research reactors or spallation sources are brought to an equilibrium temperature by collisions with a moderator, the temperature T in Eq.


19

(1.57) is the temperature of the moderator. Neutrons moderated at a few hundred degrees Celsius have wavelengths comparable to interatomic distances, and can thus be used in diffraction. The rms energy of neutrons moderated at 400 K, is, according to Eq. (1.56), equal to 8.3 • 10 -21 J, or 0.05 eV. This is comparable to the spacing of the energy levels of phonons in a crystal (see chapter 2), and the basis for phonon spectroscopy based on inelastic neutron scattering. On the other hand, the energy of 1 A photons is higher by several orders of magnitude, as can be easily verified from A photon scattered inelastically by energy exchange with a phonon therefore loses (or gains) an extremely small fraction of its energy, which can only be detected with supreme energy resolution. In the case of neutrons, the change is easily measurable by energy analysis of the diffracted beam. 1.4.2 The Neutron Scattering Length The principal contribution to neutron scattering is due to the interaction between the atomic nuclei and the neutrons (Bacon 1962, Squires 1978). Since the atomic nuclei have dimensions of 10 -13 cm (1 fermi), and nuclear forces have about this range, the nucleus acts as a point scatterer for neutrons with For practical purposes, the neutron scattering amplitude, or scattering length b, is therefore independent of unlike the X-ray form factor. Furthermore, since the scattering length is a property of the nucleus, it is different for different isotopes of the same element. For some isotopes with energy-level spacings close to the energy of thermal neutrons, scattering is a resonance phenomenon, and wavelength dependent, in analogy to anomalous X-ray scattering. Examples are 103 Rh, 113Cd, and 157Gd. For most nuclei, however, the scattering length is wavelength independent. The value of the scattering length depends also on the spin state of the compound nucleus consisting of the neutron and the scattering nucleus. As the neutron has a spin 1/2, every nucleus with nonzero spin has two values of the scattering length. None of these values can be predicted by nuclear theory, so experimentally determined values must be used. The random occurrence of different scatterers on identical crystallographic sites is equivalent to structural disorder, and leads to an incoherent component of the scattering. If the relative frequency of occurrence of the isotopes in specific spin states is fi, the average nuclear scattering length b will be given by Some values for b are listed in Table 1.1. It is the amplitude of scattering to be used in the structure factor expression for the amplitude of the Bragg reflections. In analogy to Eq. (1.26),

The incoherent scattering is given by the difference between the total scattering and the coherent scattering. For a monatomic crystal with one atom per unit cell,

20


FIG. 1.8 Coherent neutron scattering lengths for the elements in their natural abundance, in units of Fermi (1 fermi = 10- 13 cm).

The spin-incoherent scattering is especially pronounced for the hydrogen atom. Atom scattering lengths for the elements in their natural abundance are shown in Fig. 1.8, while some coherent and incoherent scattering lengths are listed in Table 1.1. It is of interest that the scattering lengths are of the same order of magnitude as the scattering amplitude of the free electron, which equals 2 . 8 1 8 - 1 0 - I 3 cm, or 2.818 fermi. The incoherent scattering length is zero for zero-spin nuclei with even numbers of protons and neutrons. As a consequence of the large incoherent values for 'H, it is important in accurate work to substitute deuterium for hydrogen whenever possible, in order to reduce the incoherent background in the diffraction pattern. The scattering lengths discussed so far refer to a fixed nucleus. If the nucleus is free to vibrate, it will recoil under the impact of the neutron. In that case the effective mass is that of the compound nucleus, consisting of the neutron and the scattering nucleus. This means that the neutron mass m must be replaced by the reduced mass of the compound nucleus where M is the mass of the scattering atom. As a result, the scattering length of the free atom is related to that of the bound atom by

The difference is usually small, except for the very light elements. For 'H, for example, the free scattering length is only half that of the bound proton.


21

T A B L E 1.1 Values of Coherent and Incoherent Scattering Lengths in Fermi (1 fermi = 10 - l3 cm) for a Number of Nuclei Abundance Nucleus

!H ?H

126c 136c

50C r 2 4 Lr 52Cr 241-' 53C r

24 Lr

54Cr 5525Mn 234

u 235u 238 u

/

(%) 99.985 0.015 98.90 1.10 4.35 83.79 9.50 2.36

100 0.005 0.72

99.275

1/2 1 0

1/2 0 0 3/2 0 5/2 0 7/2

0

bcoh

binc

-3.74

25.22

6.67 6.65 6.19 -4.50 4.92 -4.20

4.55 -3.73 12.4 10.5 8.4

4.03 0 0.52

0 0 6.86

0 1.79

0 1.3 0

Source: International Tables for Crystallography, Vol. C (1992).

The great advantage of elastic neutron scattering over X-ray scattering is that it gives information directly on the nuclear positions, without being influenced by the details of the charge density distribution. The combination of the two techniques can be used advantageously to separate the effects of thermal vibrations from the effects of chemical bonding on the X-ray scattering, as discussed further in chapter 5.

2

The Effect of Thermal Vibrations on the Intensities of the Diffracted Beams

The atoms in a crystal are vibrating with amplitudes determined by the force constants of the crystal's normal modes. This motion can never be frozen out because of the persistence of zero-point motion, and it has important consequences for the scattering intensities. Since X-ray scattering (and, to a lesser extent, neutron scattering) is a very fast process, taking place on a time scale of 10 -18 s, the photon-matter interaction time is much shorter than the period of a lattice vibration, which is of the order Thus, the recorded X-ray scattering pattern is the sum over the scattering of a large number of instantaneous states of the crystal. To an extremely good approximation, the scattering averaged over the instantaneous distributions is equivalent to the scattering of the time-averaged distribution of the scattering matter (Stewart and Feil 1980). The structure factor expression for coherent elastic Bragg scattering of X-rays may therefore be written in terms of the thermally averaged electron density:

The smearing of the electron density due to thermal vibrations reduces the intensity of the diffracted beams, except in the forward \S\ = 0 direction, for which all electrons scatter in phase, independent of their distribution. The reduction of the intensity of the Bragg peaks can be understood in terms of the diffraction pattern of a more diffuse electron distribution being more compact, due to the inverse relation between crystal and scattering space, discussed in chapter 1. The reduction in intensity due to thermal motion is accompanied by an increase in the incoherent elastic scattering, ensuring conservation of energy. In this respect, thermal motion is much like disorder, with the Bragg intensities 22

Thermal Vibrations and Diffracted-Beam Intensities

23

representing the average distribution, and the deviations from the average appearing as a continuous, though not uniform, background, generally referred to as thermal diffuse scattering or TDS. 2.1 The Normal Modes of a Crystal 2.1.1 Phonons, Internal and External modes A crystal with n atoms per unit cell has 3nN degrees of freedom, N being the number of unit cells in the crystal. Thus, subtracting the translations and rotations of the crystal as a whole, there are normal modes. Since the displacements of atoms in different cells are correlated, the normal modes are waves, or phonons, extending over the crystal, with force constants obtained from a sum over the interactions between atoms in all unit cells, and wavevector q. For a small change in magnitude of q, the change in frequency to is small, and is a continuous function of The dependence of on is referred to as the dispersion relation. The number of phonon branches with continuously varying .. equals 3n, but some of these may be degenerate due to the symmetry of the crystal. For a molecular crystal, the description can be simplified considerably by differentiating between internal and external modes. If there are M molecules in the cell, each with nM atoms, the number of external translational phonon branches will be 3M, as will the number of external rotational branches. When the molecules are linear, only 2M external rotational modes exist. For each molecule, there are 3nM — 6 (3nM — 5 for a linear molecule) internal modes, the wavelength of which is independent of q. Summing all modes gives a total number of N{M(3nM - 6) + 6M} = 3nN, as required, because each of the modes that have been constructed is a combination of the displacements of the individual atoms. In the harmonic approximation, a mode is described by a single force constant k f , equal to the ratio between the magnitudes of the displacement and the restoring force (Hooke's law). For molecular crystals, the force constants of the internal modes are usually much larger than those of the external modes. Frequencies of the former are typically in the 500-3000 c m - 1 range, similar in value to those of the isolated gas-phase molecule. Phonon frequencies tend to be very much lower, in the 20-200 cm-1 range. The variation of the phonon frequency with the wavelength of the phonon is pronounced for acoustical modes, for which all particles (molecules for a molecular crystal, and atoms otherwise) in the unit cell move in phase. If the unit cell contains more than one particle, optical modes exist, for which two particles in the same unit cell have opposite phase. The dispersion curves for the rock-salt structure are illustrated in Fig. 2.1, and for the rigid body vibrational modes of the molecular crystals anthracene and naphthalene in Fig. 2.2. 2.1.2 The Frequency of the Normal Modes The theory of lattice vibrations was developed by Born and von Karman (1912, 1913), and is described in detail in a monograph by Born and Huang (1954). The

24


equations of motion of a vibrating assembly of atoms j, each of mass m(j), are given by

in which ii is the vector of the second time-derivatives of the displacements u and is a 3 x 3 force constant matrix, with elements

where V is the potential giving rise to the forces. Thus, — is the force exerted in the direction on atom j when atom j' is given a small unit displacement in the direction. The summation in Eq. (2.2a) is over all atoms in the crystal, but, in general, the largest contributions originate from nearestneighbor interactions. The number of independent elements of may be restricted by symmetry. In the face-centered cubic structure, for example, the force constant matrix for two atoms 1/2 1/2 0 apart is given by (Willis and Pryor 1975)

in which the zero elements correspond to cases in which one of the two directions a and a' is perpendicular to the internuclear vector, and the second has a component along this vector. For harmonic oscillations the time-dependent displacements u of atom j are related to the amplitudes of vibration U(y') by in which w is the angular frequency, equal to 2nv. The amplitude U(j) is complex when the phase at t = 0 is taken into account, that is, Substituion of Eq. (2.4) into Eq. (2.2a) gives

The phonons are not stationary modes, but traveling waves extending through the whole crystal. The momentum of a phonon can be assigned as equal to hq, in analogy with the momentum of a photon, though it is not strictly defined, as the phonon can be described equivalently in an extended Brillouin zone (see Fig. 2.1), corresponding to a different value of the wavevector q. The displacements of a particle j at r in unit cell /, subject to a phonon wave with wave vector q, obey the equation


25

FIG. 2.1 Dispersion curves for the rock-salt structure. The terms m l and m 2 are the masses of the two ions in the rock-salt structure. The relative directions of displacement of the two types of atoms at a number of points along the branches are indicated. Source: Willis and Pryor (1975). Reprinted with the permission of Cambridge University Press.

Using Eq. (2.5), this becomes

Note that atoms in different unit cells have the same amplitude of vibration for a particular phonon; they differ in the phase of the traveling wave because of the q-r(jl) term. The equation of motion (2.2a) can now be rewritten as the sum over the atoms in a unit cell and the sum over all unit cells:

As before, the element

is the force in the

direction exerted on

atom (jl) when atom (j"l') is given a small unit displacement in the «' direction; the index j runs from 1 to n, the number of atoms in the unit cell, and / runs over all unit cells. It is clear from Eq. (2.2b) that the frequency w in Eq. (2.7) is a function of q, because q governs the relative displacement of two interacting atoms. The dependence on q (the dispersion relationships) is illustrated in Fig. 2.1 for the rock-salt structure. It can be shown that all normal modes can be represented in the first Brillouin zone, which extends from in the a direction of the rock-salt structure, or, more generally, is bounded by faces located halfway between the reciprocal lattice points in the space defined by 1 The Note that compared with Eq. (1.21) a factor 2n occurs on the right-hand side of this equation.

26


number of normal modes within the first Brillouin zone along each of the branches in Fig. 2.1 is equal to the number of unit cells in the crystal along the a direction. Substitution of Eq. (2.7b) into Eq. (2.2b) leads to the matrix equation

where Um is a column matrix of the mass-adjusted displacement coordinates

in which m is a 3n x 3n diagonal matrix obtained by repeating the masses three times along the diagonal (for the three displacements of each atom). Term D is the mass-adjusted dynamical matrix of the crystal, an element of which is given by

The elements of D represent the sum over all unit cells of the interaction between a pair of atoms. D has 3n x 3n elements for a specific q and j, though the numerical value of the elements will rapidly decrease as pairs of atoms at greater distances are considered. Its eigenvectors, labeled ea(j | /kq), where k is the branch index, represent the directions and relative size of the displacements of the atoms for each of the normal modes of the crystal. Eigenvector ea(j | /kq) is a column matrix with three rows for each of the n atoms in the unit cell. Because the dynamical matrix is Hermitian, the eigenvectors obey the orthonormality condition

that is, when summed over all three directions a and all atoms, the product is equal to 1 if k — k', and zero otherwise. The eigenvectors are, in general, complex, as a phase vector is included, except for an atom or molecule located on a center of symmetry, for which the phase is symmetry restricted to be zero or The mean-square displacements of each of the atoms in the crystal, which affect the the X-ray scattering amplitudes, are obtained by summation over the displacements due to all normal modes, each of which is a function of ex(j \ /cq), as further discussed in section 2.3. The eigenvalues of D are the frequencies of the normal modes. For a molecular crystal, the internal modes tend to be q independent and thus appear as horizontal lines in Fig. 2.1; n is then equal to the number of molecules M in the cell, leading to a considerable simplification. The resulting dynamical matrix has 6M x 6M elements, considering both translational and rotational motions, and atom-atom potential functions may be used for its evaluation. Dispersion curves obtained in this manner for anthracene and naphthalene, are illustrated in Fig. 2.2. The relation between the crystallographic temperature factors and the eigenvectors e of D is discussed in section 2.3.


27

FIG. 2.2 Dispersion curves for anthracene and naphthalene for external-mode phonons traveling in the b direction. In each crystal there are two molecules in the unit cell, leading to 12 external phonon branches. The librational modes are labeled R. Source: Pawley (1967).

2.2 The Effect of Thermal Vibrations on the Bragg Intensities 2.2.1 The Born-Oppenheimer Approximation

Since electrons are much lighter than nuclei, they move much faster, and may be assumed to adjust instantaneously to a change in nuclear configuration. This is the background for the Born-Oppenheimer approximation, which allows separation of the nuclear and electronic energies. In the Born-Oppenheimer approximation, the electronic energy and the electron distribution are functions of the instantaneous nuclear coordinates, the dependence on the nuclear coordinates resulting from the electrostatic attractions between the electrons and the nuclei. The time-averaged electron density is then the weighted average of the electron density for each of the nuclear configurations which occur along the nuclear vibration path, the weights being determined by the nuclear probability distribution P ( u l 5 . . . , UN), where u 1 , . . . , u N are the displacement coordinates. Thus, if is the electron density at r corresponding to the nuclear geometry u 1 , . . . , U N , the time-averaged electron density is

28


When the electrons can be assigned to specific nuclei, and follow these nuclei perfectly, the density for such a rigid group can be written as

The label of rigid group as used here may apply to atoms or rigidly connected groups of atoms. In the former case, we obtain

Expression (2.14), referred to as the convolution approximation, is widely applied in crystallographic work. 2.2.2 The Harmonic Temperature Factor According to the Fourier convolution theorem, further discussed in section 5.1.3, the Fourier transform of the convolution in expression (2.14) is the product of the Fourier transforms of the individual functions, or

Thus, the temperature factor T(S) is the Fourier transform of the probability distribution P(u): T(S) = F{P(u)}. In the common case that the rigidly vibrating groups are considered to be the individual atoms, T(S) is the Fourier transform of the atomic probability distribution. For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch's theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). For an isotropic potential, the three-dimensional probability distribution is given by 2

where is the (isotropic) mean-square displacement. The corresponding temperature factor follows from

The integral in Eq. (2.17) is the product of the integrals over each of the Cartesian

2 In practice, the probability distribution often includes static disorders in the crystal. The temperature parameter B in such cases is more properly described as a mean-square displacement parameter.

Thermal Vibrations and Diffracted-Beam Intensities 29 coordinates, / = IxIyL. For the integration over the x coordinate,

as can be verified by completing the square in the exponent of the second integral by multiplication with the x-independent term e2*s2x ... du"r). The

Einstein convention of summation over repeated indices is implied. For a distribution expanded around the equilibrium position, the first derivative is zero, and may be omitted, while the second derivatives are redundant as they merely modify the harmonic distribution. Since 0P(u) 's a Gaussian distribution, Eq. (2.28) can be simplified by use of the Tchebycheff-Hermite polynomials, often referred to simply as Hermite polynomials,3 H1....related to the derivatives of the three-dimensional Gaussian probability distribution by

The result is

3 The polynomials defined here are different from the Hermite polynomials which occur in the solutions of the Schrodinger equation for the harmonic oscillator.

32


T A B L E 2.1 Low-Order Hermite Polynomials

Note: w i = P j k x 1 ' . where the Gaussian function is defined as exp ( —1 / 2 p j k x j \ k ) .Brackets indicate averaging of the term over all permutations of the bracketed indices which produce distinct terms, noling that pit = pti and wjwk = w k - w j . Source: Johnson (1969). Johnson and Levy (1974), Zucker and Schulz (1982).

Here, the permutations of 7, k,l,... include all combinations which produce different terms. The multivariate Hermite polynomials are listed in Table 2.1 for orders < 6. Like the spherical harmonics, the Hermite polynomials form an orthogonal set of functions (Kendal and Stuart 1958, p. 156). The coefficients c in this probability distribution are referred to as the quasimoments of the distribution. Because of the orthogonality of Hermite polynomials, the quasimoments of a function are obtained by integration of the product of the function and the related Hermite polynomial over all space. For the one-dimensional case,

Thus, the quasimoments are directly related to the moments // of a distribution defined by The relations between the two sets follow by substituting the expressions for the Hermite polynomials into Eq. (2.30b). They can also be derived by writing /(x) as an expansion, both in terms of its moments [see Eq. (2.33)], and in terms of the quasimoments [as in Eq. (2.30a)], and equating equivalent terms. The Gram-Charlier temperature factor is the Fourier transform of Eq. (2.30), which is given by

where T 0 (H) is the harmonic temperature factor. As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution.


33

2.2.3.2 The Cumulant Expansion

A second statistical expansion that may be used to describe the atomic probability distribution is due to Edgeworth (Kendal and Stuart 1958, Johnson 1969). It expresses a distribution in terms of its cumulants K. If D is the differential operator, P(u) is described as

where f 0 (u) is the harmonic probability distribution. Though the differential operator in Eq. (2.32) occurs in the exponent, use of a Taylor expansion leads to an expression of a more common form. A cumulant of rank s is a symmetric tensor with (s2 + 3s + 2)/2 unique elements for a three-dimensional distribution. Like the moments and the quasimoments c, the cumulants are descriptors of the distribution. For a onedimensional distribution, the relations between the cumulants and the moments are defined by equating the two expansions:

As is evident from Eq. (2.18), the Fourier transform of an exponential is an exponential. Fourier transform of Eq. (2.32), omitting, as in Eq. (2.30), the first- and second-order terms, gives (Kendal and Stuart 1958)

Compared with the Gram-Charlier temperature factor of Eq. (2.31), the entire series now occurs in the exponent, so, in the cumulant formalism, terms are added to the exponent of the harmonic temperature factor Application of the Taylor expansion exp(i'H) = £ ( i H f / N \ to Eq. (2.34) shows that the two expressions are identical if all terms up to infinity are included. The Taylor expansion of Eq. (2.34) is

This formulation, referred to as the Edgeworth approximation (Zucker and

34


Schulz 1982), corresponds to a probability distribution which is the Taylor expansion of Eq. (2.32), and similar to the Gram-Charlier distribution of Eq. (2.30):

Relations between the cumulants Kikl and the quasimoments cjkl follow from comparison of Eqs. (2.36) and (2.30):

The result shows that the sixth- and higher-order cumulants and quasimoments differ. The third-order cumulant Kjkl contributes not only to the coefficient of Hjkl, but also to higher-order terms of the probability distribution function. The situation is analogous for cumulants of higher orders. It follows that for a finite truncation of the temperature factor defined by Eq. (2.34), the probability distribution cannot be represented by a finite number of quasimoments. This is a serious difficulty when a probability distribution is to be derived from an experimental temperature factor of the cumulant type. A second complication in the use of the cumulant expansion, pointed out by Scheringer (1985), is that the probability function always has some physically unrealistic negative regions. In certain cases of large anharmonicity, the Fourier transform of Eq. (2.34) may, in fact, not exist at all. As a result of such considerations, the Gram-Charlier expression is generally preferred over the cumulant expansion, because its truncation is equivalent in real and reciprocal space, and it does not lead to negative regions in the probability distribution (Kuhs 1983,1992; Scheringer 1985). 2.2.3.3 The One-Particle Potential (OPP) Model Unlike the Gram-Charlier and cumulant formalisms, the OPP model has a physical rather than a statistical basis. It assumes that each atom vibrates in a potential well V(u), determined by the interaction with the other atoms in the crystal, without any correlation between vibrations of adjacent atoms. In the classical high-temperature limit, kBT» hv, where kB is the Boltzmann constant, and hv is the spacing of the quantum-mechanical harmonic oscillator energy levels. If this condition is fulfilled, the energy levels may be considered as continuous, and Boltzmann statistics apply. The corresponding distribution is with N, the normalization constant, defined by

P(u) du = 1.


35

The potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates (Dawson 1967, Willis 1969) As in the probability distributions of Eqs. (2.30) and (2.32), the first derivatives vanish at the equilibrium position, so aj = 0. The constant V0 term does not affect the probability distribution, and may also be omitted. Substitution of Eq. (2.39) in Eq. (2.38), and use of the approximation exp ( — A) = 1 — A for the higher-order terms, leads to in which /?' = p/(kBT), etc. In the description, the higher-order terms appear as corrections to the harmonic temperature factor. We note that the normalization factor N depends on the level of truncation of the series. The Fourier transform of the OPP distribution, in a general coordinate system, is (Johnson 1970, Scheringer 1985)

where T0 is the harmonic temperature factor, and G represents the Hermite polynomials in reciprocal space. The harmonic term rn(H) of Eq. (2.41) is equal to Comparison with Eq. (2.23) shows that the classical OPP model predicts the elements of the anisotropic harmonic temperature parameter to be proportional to the absolute temperature. Expressions simpler than Eq. (2.41) are obtained if the OPP temperature factor is expanded in the coordinate system which diagonalizes fijk. In that case, the Hermite polynomials become products of the displacement coordinates uj (Coppens 1980, Tanaka and Marumo 1983). The first terms in the expansion are given by

This result indicates that, according to the OPP model, the higher-order anharmonic terms have a stronger temperature dependence than the leading harmonic terms. The quartic terms, not specifically included in Eq. (2.43), have an even larger temperature dependence proportional to T3. Thus, the effect of

36


anharmonicity may be effectively reduced by cooling when the harmonic temperature factor is the leading term in the expansion. This is an important result for charge density analysis, especially for heavier atoms, for which anharmonicity affecting all electron shells may mask the bonding effects of the valence-electron distribution. 2.2.3.4 Application to Diamond-Type Structures Dawson and coworkers pioneered the application of the OPP model to diamondtype structures (Dawson 1967, Dawson et al. 1967). In the diamond-type structure, common to diamond, silicon, and germanium, the atoms are located at 1/8, 1/8, 1/8, at the center-of-symmetry related position at — 1/8, —1/8, — 1/8, and repeated in a face-centered arrangement. The tetrahedral symmetry of the atomic sites greatly limits the allowed coefficients in the expansion of Eq. (2.39). With x, y, z expressed relative to the nuclear position, the potential is given by

with a corresponding OPP temperature factor, obtained by omitting the symmetryforbidden terms from Eq. (2.43),

When the atomic scattering factor is real (as it is when bonding effects on the charge density are neglected), and resonance scattering has been corrected for, the harmonic structure factor expression is equal to

where T0 represents the harmonic temperature factor, defined by the exponential factor in Eq. (2.45). Thus, in this approximation, F(H) equals zero for reflections with h + k + I = 4n + 2, such as (222), (442), and (662). However, the atoms vibrate more strongly into the void opposite the tetrahedrally arranged bonds, while the displacement in opposite directions into the bonds is more constrained. As a result, -/ in Eqs. (2.44) and (2.45) has a significant negative value, leading to nonzero intensity due to anharmonicity for reflections with h + k + l = 4n + 2 and large values of the product hkl. The corresponding intensity increases with temperature, because the anharmonic term is proportional to T2. This effect, and its relation to the scattering of the covalent bonding density, is further discussed in chapter 11. 2.2.4 Comparison of the Anharmonic Formalisms The OPP formalism, though based on the assumption of independent motion, has the advantage of assigning a physical meaning to the terms in the expansion. By equating the OPP terms to the corresponding ones in the statistical expansions, the quasimoments and cumulants can be related to the parameters of the potential model, and their temperature dependence can be predicted.


37

FIG. 2.4(a) The coordination of Al(4) in VA1 10.42 . Thermal ellipsoids (100 K) are at the 50% probability level. Four nearest-neighbor Al(3) atoms are located 3.1330 (4) A from Al(4) in the [111] directions, while 12 Al(l) atoms are at 3.1484 (5) A, arranged in groups of three, capping each face of the tetrahedron formed by the Al(3) atoms. Source: Kontio and Stevens (1982).

For a cubic site, relations between the cumulants and the coefficients of the OPP model have been derived by Kontio and Stevens (1982), and applied to the Al(4) atom in the alloy VA110. 42. The coordination of Al(4) is illustrated in Fig. 2.4(a), while the potential along [111], derived from the thermal parameter refinement, is shown in Fig. 2.4(b). It is clear from these figures that higher than third-order terms contribute to the potential, because the deviation from the harmonic curve is not exactly antisymmetric with respect to the equilibrium configuration. The potential appears steeper at the higher temperature, which is opposite to what is expected on the basis of the thermal expansion of the solid. 2.2.5 Quantum-Statistical Treatments The classical model predicts thermal motion to vanish at very low temperatures, in contradiction to the zero-point vibrations which follow from the quantummechanical treatment of oscillators. For temperatures at which hv K kBT, the spacing of the discrete energy levels cannot be neglected, so the classical model is no longer valid. For a harmonic oscillator with frequency v, the potential energy equals l/2/Cy, where k f is the force constant. Using the virial theorem, which states that £ kin = £pol = E tot /2,

38


FIG. 2.4(b) The experimental potential along the [111] direction for the Al(4) atom in the alloy VA1 1 0 4 2 , obtained with the one-particle potential temperature factor. Left: 100 K; right: room temperature. The broken lines represent the harmonic components. Differences between the classical and quantum-statistical results at both temperatures were found to be extremely small. Source: Kontio and Stevens (1982).

According to statistical mechanical theory, the total energy of the oscillator is a function of the partition function z, through the relation

where T is the absolute temperature. The partition function z for the harmonic oscillator is obtained by the summation over all levels £ = vhv, where v is the vibrational quantum number, and E is counted from the lowest (v = 0) level, at hv/2. The well-known result is

Substitution into Eq. (2.48), and adding the zero-point energy hv/2, gives

Combining this result with Eq. (2.47), and substitution of k — 4n2mv, where m is the reduced mass of the harmonic oscillator, gives the quantum-statistical temperature dependence of as

or, equivalently,


39

FIG. 2.5. Temperature dependence of the mean-square (ms) displacement of a quantummechanical harmonic oscillator with a mass of 200 daltons for a number of frequencies. The linearity of the ms displacement in the high-temperature region is evident.

This expression is illustrated in Fig. 2.5 for a number of frequencies. For large T, with e4 = 1 + A, this reduces to

Thus, in the high-temperature limit, the mean-square displacement of the harmonic oscillator, and therefore the temperature factor B, is proportional to the temperature, and inversely proportional to the frequency of the oscillator, in agreement with Eq. (2.43). At very low temperatures, the second term in Eq. (2.5la) becomes negligible. The mean-square amplitude of vibrations is then a constant, as required by quantum-mechanical theory, and evident in Fig. 2.5. Quantum-statistical expressions such as Eq. (2.51) can be generalized for anharmonic motion, as shown by Mair (1980). 2.2.5.1 Numerical Application of Expression (2.51) Substitution of values for the physical constants yields the numerical expression

40


The temperature dependence according to this expression is illustrated in Fig. 2.5 for a number of frequencies, and a mass of 200 daltons. As the mean-square displacement scales with the inverse mass of the oscillator, the onset of the linear region is independent of mass for a given frequency. As at very low temperatures, for very high frequencies the thermal motion becomes temperature independent. For example, for the C—H stretching mode, with frequency in the 2700-3300 c m - 1 range, the exponential in the denominator of Eq. (2.51) is very large for common temperatures and the second term in the square brackets is negligible. Using for m the reduced mass of the oscillator (0.9231 dalton for diatomic C—H) gives, with v = 3000cm - 1 , a constant mean-square vibrational amplitude of 0.006 A2. Vibrations in a real crystal are described by the lattice dynamical theory, discussed in section 2.1, rather than by the atomic oscillator model. Each harmonic phonon mode with branch index k and wavevector q then has, analogous to Eq. (2.50), an energy given by

2.3 The Relation between the Atomic Temperature Factors and Lattice Dynamics 2.3.1 General Expression As the oscillators of the OPP model vibrate independently of each other, the frequencies are dispersionless, that is, independent of a wavevector q. For the internal modes of a molecular crystal, this tends to be a very good approximation. For the external modes, the dispersion can be pronounced, as shown in Figs. 2.1 and 2.2. In order to obtain the mean-square vibrational amplitudes for the latter, a summation over all phonon branches in the Brillouin zone must be performed. The atomic displacements for the mode (/cq) are described by the 3n column matrix U0, which we used in Eq. (2.9). Matrix U is a function of the eigenvectors e(/cq) of the mode, and its amplitude A(kq):

The displacements are related to the energy of a mode by the expression

where N is the number of unit cells. This expression is analogous to Eq. (2.47), EJ = kj , for the single harmonic oscillator, as can be verified by the substitution of kf = imo2 in the latter expression. The displacement of an atom j, in unit cell /, at time f, is obtained by the summation over all 3nN normal modes, combined into 3n phonon branches,


41

or, using Eq. (2.55),

The tensor j, describing the mean-square displacements of atom j, is the time average (T) (the presuperscript denotes the axis that is shifted; the subscript denotes the direction of the shift component). Analogous equations for displacements of the L2 and L3 axes are obtained by permutation of the indices. If all three axes are appropriately displaced, only the diagonal terms of S remain. Referred to the principal axes of L, they represent screw correlations along these axes and are independent of origin shifts. The elements of the reduced translation tensor T are:

and

The resulting description of the average rigid-body motion is in terms of six independently distributed instantaneous motions—three screw librations about nonintersecting axes (with screw pitches given by S11/Lli, etc.) and three translations. The parameter set consists of three libration and three translation amplitudes; six angles of orientation for the principal axes of L and T; six coordinates of axis displacement; and three screw pitches, one of which has to be chosen arbitrarily; again, for a total of 20 variables.


47

TABLE 2.3 Rigid-Bond Test for p-Nitropyridine-JV-oxide at 30 K

Spherical-Atom Refinement (7(w2) ~ 0.0003 A2 A

B

N(4)

C(6) C(7)

C(6) C(7) C(8) C(9) C(IO) N(4) C(8) N(5) N(5)

C(8) C(9) C(10) N(4) 0(1) N(5) 0(2) 0(3)

rms discrepancy

Charge-Density Model — 1, /< > 0), with n = n, + 2 (integration over three dimensions).

Chemical Bonding and the X-ray Scattering Formalism

65

T A B L E 3.4 Best Sinj;le C Dens ity Valiies (Bohr" 1 ) f or the Val ence Shell s of a f dumber of Atoms

Carbon Nitrogen Oxygen Fluorine

Silicon

2s

2p

3.216

3.136

3.947 4.492

3.934 4.453

5.128

5.110

3s

3p

3.269

2.857

Average sV 3.176 sy 3.940 sY 4.469

*V

5.115

Average

*V

3.063

Source: Clement! and Raimondi (1963).

Energy-optimized, single-Slater £ values for the electron subshells of isolated atoms have been calculated by dementi and Raimondi (1963). For the electron density functions, such ( values are to be multiplied by a factor of 2. Values for a number of common atoms are listed in Table 3.4, together with averages over electron shells, which are suitable as starting points in a least-squares refinement in which the exponents are subsequently adjusted by variation of K'. A full list of the single £ values of Clementi and Raimondi can be found in appendix F. The coefficients n, have to obey the condition n, ^ /, imposed by Poisson's electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). Similarly, octupoles and hexadecapoles can be thought of as arising from 2p3d and 3d3d atomic orbital products, which leads to n, = 3 and 4, respectively. The

FIG. 3.6 Illustration of the product of two spherical harmonic functions: pPx-py

= dxy

66

X - r a y Charge Densities and Chemical Bonding T A B L E 3.5 Position of Radial M axima (A) 1for Different n,, (K = 1 ami ( Values 0.65 A"1 appears adequate in the cases studied. Improved scattering models, as discussed in chapter 3, are an effective means of reducing scale factor bias, though accurate experimental k values remain preferable. That the positive bias in the scale factors correlates with an increase in thermal parameters is evident from comparison of X-ray and neutron results (Coppens 1968). The apparent increase in thermal parameters of some of the atoms may be interpreted as the response of the spherical-atom model to the existence of overlap density. Because of the positive correlation between the temperature parameters and k, this increase is accompanied by a positive bias in k. 4.3 Physical Constraints of the Electron Density 4.3.1 The Electroneutrality Constraint Since a crystal is neutral, the total electron population must equal the sum of the nuclear charges of the constituent atoms, or

where Zf is the nuclear charge of the ith atom. Equivalently, starting from a neutral crystal,

where dPLj are the calculated shifts for population parameter; on atom i, and qtj is the charge integrated over the corresponding density function. With the normalizations discussed in chapter 3, qtj will be either one for I = 0 or zero for I ^ 0. Thus, Eq. (4.37a) can be rewritten as

Several ways to introduce the constraint into the refinement are discussed in the following sections. 4.3.1.1 The Hamilton Method In a convenient method, due to Hamilton (1964), the Lagrangian multipliers representing the constraint are algebraically eliminated from the least-squares expressions. The linear constraints are defined as where values of X are the parameter shifts dxh Qbm is a coefficient matrix, and the column vector Z M contains the values to which the linear combinations are constrained. Hamilton has shown that the best constrained estimates X are

84


obtained from the shifts without the constraints, using the equation

In the case of the electroneutrality constraint, Z in Eq. (4.38) equals 0, so that

where Q is a row vector with a 1 for every element representing the population of a normalized monopole, and 0 otherwise. The variance-covariance matrix with the constraint is modified in a similar manner, accordine to the exoression Equations (4.40) and (4.41) are easily implemented in an existing least-squares program and give both the constrained and the unconstrained results in a single refinement cycle. However, the method fails if the unconstrained refinement corresponds to a singular matrix, as would be the case, for example, if all population parameters, including those of the core functions, were to be refined in addition to the scale factor k. 4.3.1.2 The Use of Independent Variables The introduction of dependent variables is an often-used approach in handling constraints. It is, for example, quite common in crystals that one site is occupied by two different atoms, or one atom is distributed over two sites. In this case, a single population parameter P^ may be introduced, such that Pl = P\, and P2 = 1 - Pi. More generally, the shifts in the new set of independent variables Y are related to the shifts in the dependent variables X by (Raymond 1972) The matrix J represents the constraints. If there are b constraints, J will be an m x (m — b) matrix, where m is the number of variables before the introduction of the constraint. The derivatives of the observations with respect to the independent parameters Y are obtained from

For example, in a refinement of three population parameters, satisfying the constraint £ Pj = nc, P: and P2 may be chosen as the set Y. Equation (4.42) then becomes

The independent-variable method leads to a reduction in the size of the matrix B from m x m to (m — b) x (m — b).

Least-Squares Methods in Charge Density Analysis

85

4.3.1.3 The F(OOO) Constraint Electroneutrality may also be implemented by imposing the requirement that F(OOO) equal the number of electrons in the unit cell. The equation F(OOO) = nt can be treated as an observation, with a weight sufficient to keep the crystal practically neutral, but sufficiently small such as not to dominate the least-squares treatment. This slack constraint (Pawley 1972) has been applied in electron density analysis by Hirshfeld (1977). 4.3.1.4 The Use of a Core Scale Factor When the core populations are not varied, a somewhat different approach can be used, in which the scale factor k multiplies only the functions with fixed electron population, rather than all the electron functions (Stewart 1976). The observational eauations are of the form

The valence-shell populations are scaled after completion of the refinement by use of FfJ = P-j/k. If the procedure is successful, the sum of PtJ should be close to the total number of valence electrons. The difference between these two quantities is used as a test of the adequacy of the valence density functions. 4.3.2 The Hellmann-Feynman Constraint The electrostatic Hellmann-Feynman theorem states that for an exact electron wave function, and also of the Hartree-Fock wave function, the total quantummechanical force on an atomic nucleus is the same as that exerted classically by the electron density and the other nuclei in the system (Feynman 1939, Levine 1983). The theorem thus implies that the forces on the nuclei are fully determined once the charge distribution is known. As the forces on the nuclei must vanish for a nuclear configuration which is in equilibrium, a constraint may be introduced in the X-ray refinement procedure to ensure that the Hellmann-Feynman force balance is obeyed (Schwarzenbach and Lewis 1982). In a bond A—B between first-row atoms, the internuclear repulsion is electrostatically balanced by the buildup of charge density in the bond, that is due to either a- or n-valence electrons (Hirshfeld and Rzotkiewicz 1974). But this charge migration is accompanied by a polarization of the cr-valence electrons within the 2s radial nodal surface, and, through exchange interaction between the a electrons and the core, by a very sharp polarization of the core electron density in the regions very close to the nuclei. Such deformations correspond to density functions with large values of the orbital exponent coefficient £. Functions of this type are normally not included in the X-ray model, because limited resolution and thermal smearing hamper determination of their parameters. Since they are located very close to the nuclei, the occupancy of the polarization functions correlates strongly with the atomic coordinates. The Hellmann-Feynman force-balance

86


constraint indirectly provides the missing information. Enforcing the constraint will adjust the parameters such as to maintain force balance. The Hellmann Feynman constraint has been applied successfully to the exocyclic fluorine, carbon, and nitrogen atoms in tetrafluoroterephthalonitrile (1,4-dicyano-2,3,5,6-tetrafluorobenzene). Charge balance is achieved without deterioration of the least-squares agreement factors, though the resulting changes in the density maps are very small (Hirshfeld 1984) (see chapter 5). We note that the constraint applies to the static density. Its application therefore requires adequate deconvolution of thermal motion and electron density effects. 4.4 Joint Refinement of X-ray and Neutron Data 4.4.1 The Use of Complementary Information The use of complementary data from different techniques can be a powerful tool to reduce parameter correlation in least-squares methods and to obtain the best estimates compatible with all available physical information. The joint use of X-ray and neutron diffraction data is particularly expedient. Firstly, the interaction between the magnetic moments of neutrons and electrons is the basis for polarized-neutron diffraction, from which the unpaired spin density in a system can be derived. The diffraction of spin-polarized neutrons is an important technique, beyond the scope of this volume. Secondly, the interaction between neutrons and the atomic nuclei, which is the basis for structure determination by neutron diffraction, leads directly to information on the positions and mean-square vibrations of the nuclei. Neutron diffraction is especially important for the location of hydrogen atoms, as the pronounced effect of bonding on the hydrogen-atom charge density leads to a systematic bias in the X-ray positions, as discussed in chapter 3. If the charge density in a hydrogen-containing molecule is to be studied, independent information on positions and thermal vibrations of the H atoms is invaluable. The combination of data from different techniques raises several issues, as the experimental conditions may not have been identical, and the data no longer form a homogeneous set of observations. Several of these issues are discussed below. 4.4.2 Differences in Temperature Parameters Early studies, which did not include many high-order reflections, revealed systematic differences between spherical-atom X-ray- and neutron-temperature factors (Coppens 1968). Though the spherical-atom approximation of the X-ray treatment is an important contributor to such discrepancies, differences in data-collection temperature (for studies at nonambient temperatures) and systematic errors due to other effects cannot be ignored. For instance, thermal diffuse scattering (TDS) is different for neutrons and X-rays. As the effect of TDS on the Bragg intensities can be mimicked by adjustment of the thermal parameters, systematic differences may occur. Furthermore, since neutron samples must be

Least-Squares Methods in Charge Density Analysis

87

larger, because the beams are weaker, extinction and multiple diffraction tend to be more pronounced. On the other hand, absorption of neutrons within the crystal is generally lower, notwithstanding the larger specimen sizes. Such effects will contribute to the discrepancies between X-ray (X) and neutron (N) temperature parameters, which have been found to exist even room-temperature studies, for which temperature ambiguities are minimal (Craven and McMullan 1979). The effect of TDS can be especially pronounced in room-temperature studies of often soft molecular crystals, and, if not recognized, can lead to an artificial enhancement of features in difference maps based on a combination of the two techniques (Scheringer et al. 1978) (see chapter 5). To account for temperature factor differences, a temperature scale factor kT multiplying the neutron temperature parameters may be introduced, as defined by the expression (Coppens et al. 1981)

This formulation is appropriate in the "high-temperature limit" at which temperature factors are proportional to absolute temperature (chapter 2). For most molecular crystals, this limit is reached even at liquid-nitrogen temperature. Other alternatives exist, such as,

where Al/^, common to all atoms, may be chosen either to follow the symmetry transformation of Vit for symmetry-equivalent atoms, or to be similarly oriented for all atoms in the unit cell. The latter choice would correct for systematic errors due to incorrect allowance for absorption or anisotropic extinction in either of the two data sets. Blessing (1995) has tested a number of alternative models for describing the discrepancy between X-ray and neutron thermal vibration parameters, using X-ray and neutron U^ values from the lUCr oxalic acid project (section 12.1.3, Coppens 1984). The expressions UIJk,x= Uijk,N + At/y [i.e., Eq. (4.47)] and UlJk.x = kTUlJll,N + AUjj [a combination of Eqs. (4.46) and (4.47)] were found to be the most effective in providing reliable corrections. Blessing points out that the corrections calculated for the nonhydrogen atoms of a crystal can be used for adjusting neutron (7,/s of the hydrogen atoms, to provide a set of fixed hydrogen thermal parameters for use in an X-ray analysis of the charge density. 4.4.3 Relative Weighting of the X-ray and Neutron Data Because the error function

is a weighted sum of the squared

discrepancies between observed and calculated values, the relative weighting of

88


the two data sets affects their relative importance in the minimalization procedure. Since the weights should be based on the experimental uncertainties, they may be derived from the agreement between symmetry-related reflections in each of the data sets, and, counting statistical considerations, by using expressions described in the literature (McCandlish et al. 1975). 4.4.4 Estimate of the Goodness of Fit The goodness of fit achieved in a refinement is defined by Eq. (4.27). Its evaluation for each of the subsets of data requires partitioning the unknowns between the two sets. Some, such as the scale factors, will be dependent on one subset only; other, such as heavy-atom positional parameters, will be determined by both sets of data. Information on the relative dependency is contained in the matrix B, the elements of which are the sums of the contributions from the X-ray and neutron observations. We may write B = Ex + BN (4.48) For each unknown Uj the fractional dependence (p can be based on the relative contributions of the two subsets to the diagonal elements of B, according to (Coppens et al. 1981)

where E is either F or F2 depending on the function minimized, and (pxj + (pNj = 1. In case a diagonal approximation to B is used in the least-squares refinement, (px and x from a spherical-atom refinement and 0.4)

A/

3.7-4-0 (1.0) 3.17 3.6 (2.0)

Source: Ciao et al. (1992), Frost-Jensen et a). (1995).

interpretation of the spectroscopic results in terms of the detailed charge distribution, and can provide the signs, which are not generally available via the other methods. For transition metal atoms, the dominant contribution to the electric field gradient at a nucleus originates in the valence shell centered on that nucleus. The application of Eq. (8.41) to transition metal complexes will be discussed in chapter 10. Peripheral contributions become important when short interatomic distances are involved, as, for example, for the EFG at nitrogen nuclei and especially at nuclei of hydrogen atoms. Since hydrogen has only one electron, the electric field gradient is mainly due to the density farther from the nucleus, and has therefore been described as less sensitive to the precise charge distribution (Tegenfeldt and Hermansson 1985). This is borne out by calculations, and by the good agreement between X-ray, and NMR, and theoretical values, illustrated for the molecule of benzene in Table 8.3. Unless the hydrogen atoms participate in strong hydrogen-bonds, such as in T A B L E 8 .3 Experimental and Theoretic:al Values of the Elecltrie Field Gradieint at the Deuterons in Deuterolbenzene (e au ~ 3) X-ray"

V£« VEn VE., 1

0.16 (2) 0.13 (2) -0.29 (2) 0.10(1)

NMR" 0.143 0.132 -0.275 0.04

Theory (6-31G**) 0.181 0.157 -0.338 0.07

Source: ' Z Su and P. Coppens, unpublished, bas(:d on X-ray and neutron da ta of Jeffrey et al. (19!87) and private coimmunication. " Millet ami Dailey (1972).

186


LiOH -2H 2 O, the unperturbed spherical atom contribution to the EFG dominates, while the contribution from the deformation density is relatively small. The EFG data from the multipole parameters are, in principle, for the static crystal; while the spectroscopic data are affected by vibrations. There may therefore by a systematic difference between the two sets of values, which is evident for a number of hydrogen-bonded hydroxyl groups and water molecules studied by Tegenfeldt and Hermansson (1985), but is not apparent in the data in Table 8.3. The EFG values for H atoms in hydrogen-bonds is further discussed in chapter 12. 8.5 The Electrostatic Potential Outside a Charge Distribution 8.5.1 The Electrostatic Potential Outside a Charge Distribution in Terms of the Multipole Moments The potential outside a charge distribution can be expressed in terms of a finite series of the outer moments of the distribution. The expression is obtained through a power series expansion of r~l, where r is the distance from the field point to the origin of the distribution, and subsequent integration (Hirshfelder et al. 1954, Buckingham 1978). At a point r,, with components r^,, for unit value of 4ne0, one obtains

where the Einstein convention implying summation over repeated indices is used. In this expression, the moments ®xf and fia/)), of the total charge density are traceless, as defined in Eq. (7.2). Since the traceless electrostatic moments are not dependent on the spherical components of the neutral atom, they can equally well be calculated from the deformation density. The summation in Eq. (8.53) is slowly converging if a molecular charge distribution is represented by a single set of moments. However, the expression can be written as the summation over the distributed moments, centered at the nuclei j, which is precisely the information available from the multipole analysis:

in which rtj measures the distance from each center j of the expansion to the field point /'. Equation (8.54) is the limiting case of Eq. (8.49) for very large values of RMP, or exp ( — RMPZ) % 1 in the integrals ANi(,j2.k(Z, R) occurring in Eq. (8.49). 8.5.2 Net Atomic Charges Reproducing the Electrostatic Potential For a study of nonbonded interactions, it is of practical importance to identify the net atomic charges that best reproduce the potential on the periphery of the

X-ray Diffraction and the Electrostatic Potential

187

T A B L E 8.4 Diipole Moments (in Debye), Accordirig to the SCF WEive Function, from Potential-Adji asted Charges and from Mulliken C harges Derived fr om the Same Wave Function, Cornpared with Exper imental Values

H2O NH 3 N 2 CO CH 3 OH H 2 NCHO

SCF (6-31G**)

Potential Charges

2.196 1.886 2.749

2.258

1.914

4.486

Mulliken Charges 1.011

1.850

0.762

1.864

1.644 1.336 2.412

2.695 4.458

Experimental Values (Gas Phase) 1.850

1.470

2.330 1.700

3.730

Source: Chirlian a nd Francl(1987).

molecule. The idea of fitting the charges to the theoretical potential around the molecule was first proposed by Momany (1978), and elaborated by Chirlian and Francl (1987). The resulting net charges are reasonable and reproduce the molecular dipole moments from the same calculations much better than Mulliken charges, as illustrated in Table 8.4. The methods adjust the atomic net charges qt in a least-squares minimization with a discrepancy function equal to the sum of the potential differences over all n sampling points:

The location of the sampling points used in Eq. (8.55) is crucial. Typically, points are selected on and just outside the van der Waals envelope of the molecule. Woods et al. (1990) selected 100 points per atom randomly within a spherical shell around each atom, but outside the van der Waals envelope of the molecule. Ghermani et al. (1993), in an analysis of experimental charge densities of peptides and pseudopeptides (modified peptide molecules), place the sample points on the surfaces of atom-centered spheres. In the application to the pseudopeptide JV-acetyl-a,/?-dehydrophenylalanine methylamide, the sampling points are selected to be equidistant and are located on composite spherical surfaces with radii 2-8 A around the nuclei in the molecule (Fig. 8.5). The potential used as the starting point in the fitting was calculated for a molecule composed of spherical atoms, with experimental charges and contraction parameters from the ^-refinement. For a sphere radius of 2 A, the fitted charges differ by several percent from those obtained with larger radii, but convergence is reached rapidly when the sphere radius is increased, or more spheres with large radii are added. The charges obtained are within 0.06 e of the values from the /c-refinement providing the starting density for the minimalization. No molecular neutrality constraint was used; a total charge

of -0.19 e resulted. The observed and fitted potentials in the region of a peptide link are compared in Fig. 8.6. Except for the inner regions, the agreement is satisfactory. In particular, the pronounced negative areas in the proximity of the oxygen atom are well reproduced.

188


FIG. 8.5 Sampling-point shells used to obtain potential-adjusted charges of N-acetyl-a,/?dehydrophenylalanine. Source: Ghermani et al. (1993).

A geodesic scheme for the selection of sampling points has been proposed by Spackman (1996). The points are arranged on a series of fused-spherical van der Waals surfaces, and are arranged in a regular pattern, based on the tesselation of the icosahedron. The resulting charges are reported to have less dependence on molecular conformation than those from a number of other sampling techniques. Francl et al. (1996) examined the conditioning of the least squares matrix in the fitting procedure, and conclude that the method cannot be used to assign statistically valid charges to all atoms in a given molecule. This problem cannot be alleviated by the selection of more sampling points, and thus may require the introduction of chemical constraints to reduce the number of charges to be determined. Rather than fit the potential at the periphery of the molecule, Su (1993) has fitted $ at the nuclear positions. The potential at the nuclear positions can be directly related to the total energy of a molecule (chapter 9) (Polizer 1981), as well as to the atomic ionization energies (as discussed in section 8.4).

T A B L E 8.5 Comparison of Equivalent-P otential and Refinement Charges for Atoms of COO" Groups in L-Alanine (top line) and D,L-Histidine (bottom line) Potential

Refinement

Periphery

At nuclei

+ 0.55 + 0.11 -0.65 -0.68 -0.76

-0.06 -0.08 -0.33 -0.34 -0.40 -0.35

C

0(1)

0(2)

-0.65 Stutn't-': Su (1993).

K

+ 0.49 (4) + 0.51 (5) -0.58 (3)

-0.65(4)

-0.69 (3) -0.68 (4)

Multipole" -0.06(6) -0.16(22) -0.41 -0.45 -0.45 -0.48

(3) (8) (3) (9)

FIG. 8.6 Electrostatic potential maps in the region of one of the peptide links in N-acetyl-a,/?-dehydrophenylalanine methylamide. (a) Observed. (b) From net charges fitted to the potential. Contours are at 0.05 eA~ ' (1 eA)~' = 332.1 kcal mol~ '). Zero and negative contours are dashed lines. Source: Ghermani et al. (1993).

T A B L E 8.6 Molecular Dipole Moments (D) from Fit to the Potential at the Nuclei, and Directly from X-ray Refinements at nucleus 1.0

L-Alanine D,L-Histidine

MNA b PDM b 3

8.6

13.9 19.2 7.0

KM" =

1.4

12.7 16.0 23.4

8.5

K

Multipole

Refinement

Refinement

12.5 16.9 25 11

(2) (7) (6) (2)

13.3 23 (1) 25 (9) 9.8

K Expansion-con traction paraimeter in defiinition of hydrogen derisity function, MNA = 2-methy]l-4-nitroaniliile, PDM = f >yridium dicyanomethy lide. c ln H 2 O. d In dioxane. Source: Su (1993). b

Reference Li (1989) Li (1989) Howards et al. (1992) Baert et al. (1982)

Solution Value 12.3-17.0C

13.5= 6.98d 9.2"

Reference McClellan (1974) Khanarian and Moore (1980) McClellan (1963) Treiner et al. (1964)

X-ray Diffraction and the Electrostatic Potential

191

The charges obtained with nuclear sampling points are very different from those derived with the fit at the periphery. In Table 8.5, the results of the two methods at the carbon and oxygen atoms in the carboxyl groups in alanine and histidine are compared with the primary X-ray charges. Interestingly, for this example, the periphery-adjusted charges agree well with the results of the K-refinement, while the nucleus-adjusted charges are closer to those from the multipole refinement. The molecular dipole moments of a number of molecules are quite well reproduced by the nucleus-adjusted charges, particularly when the fixed K-value of 1.4 is used for the hydrogen valence shell in the fitting procedure (Table 8.6). In all fitting procedures, electroneutrality and electrostatic-moment constraints may be introduced to provide additional observational equations.

9

The Electron Density and the Lattice Energy of Crystals

9.1 The Total Energy of a System

The total energy of a quantum-mechanical system can be written as the sum of its kinetic energy T, Coulombic energy £ C oui> and exchange and electron correlation contributions E x and £ corr , respectively: The only term in this expression that can be derived directly from the charge distribution is the Coulombic energy. It consists of nucleus-nucleus repulsion, nucleus-electron attraction, and electron-electron repulsion terms. For a medium of unit dielectric constant,

where Z, is the nuclear charge for an atom at position R,, and Rfj = R, — R;. Even though the other terms in Eq. (9.1) cannot be directly calculated from the electron distribution, they can be related to it through the expressions of density functional theory, as discussed in the following sections. 9.2 Density Functional Expressions for the Energy 9.2.1 The Hohenberg-Kohn Theorem

A theorem due to Hohenberg and Kohn points to the central role of the electron density in representing the properties of a system. In 1964, Hohenberg and Kohn (1964) proved that the properties of a system with a nondegenerate ground state are unique functionals of the electron density. 192

Electron Density and Lattice Energy of Crystals

193

The proof of the Hohenberg-Kohn theorem is quite straightforward. Excluding nucleus-nucleus interactions and the nuclear kinetic energy, the Hamiltonian may be written as

where the sum is over all the electrons in the system. The term F(r,-) represents the electrostatic electron-nucleus interaction operator, while the first and the last terms are the electronic kinetic energy and electron-electron repulsion operators, respectively. Suppose a Hamiltonian H has an exact nondegenerate ground state P, with energy E, and a second Hamiltonian H' has a nondegenerate ground state T' with energy E', where H and H' differ by their local potentials V(r) and V'(r), respectively. Then *P' will not be an eigenfunction of H as long as V(r) — V'(r) is not a constant. So, if T' is used as a trial wave function for H, the corresponding energy will, according to the variational theorem be larger than the true energy. Thus, from H\l/' > Eij/'

Similarly, if T is used as a trial function of //',

In the case that p(r) = p'(r), addition of the two equations gives Since this is a contradiction, it follows that p(r) =£ p'(r). Thus, there is a one-to-one correspondence between the local potential V(r) and the electron density p(r). This implies that V, T, and £ are uniquely determined by the electron density, and therefore are functionals of the electron density. If £ is a unique functional of the electron density p, the kinetic and exchange-correlation energies, T and Exc must be functionals of p also. The Hohenberg-Kohn theorem does not go beyond this point; it offers no guidance on the nature of the functionals that it shows must exist. 9.2.2 Density Functional Expressions Density functionals are discussed extensively in the literature (Dahl and Avery 1984, Parr and Yang 1989, Ziegler 1991), and their development is an active field of research. In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the "nearly free electron gas"). In this approximation, the kinetic energy T is given by

194


where ck = ^(37r2)2/3, and the function t(p) = p(r)2/3. The exchange-correlation energy Exc is also a functional of p:

where cx = I(3/7r) 1/3 and exc(p) = p(r) I/3 . There are extensive discussions in the literature concerning to what extent Eq. (9.8) includes at least part of the correlation energy. A critical examination has been made by Sabin and Trickey (1984). Expressions (9.7) and (9.8) are referred to as local density approximations (LDA), as the functionals depend on the local density only. Nonlocal functionals include the gradient of p(r) and generally give improved agreement with exact values. Density functional calculations of molecules, using a Hamiltonian including density functionals, frequently reproduce observed properties, such as bond and excitation energies, reaction profiles, and ionization energies (Ziegler 1991). For tetrafluoroterephthalonitrile (l,4-dicyano-2,3,5,6 tetrafluorobenzene), there is excellent agreement between the electron density from a density functional calculation (Delley 1986) and the X-ray diffraction results (Hirshfeld 1992) (see chapter 5). A very et al. (1984) have proposed the use of experimental densities in crystals as a basis for band structure calculations. 9.2.3 The Total Energy as a Function of the Electrostatic Potential at the Nuclear Position In the Thomas-Fermi theory (March 1957), the electrostatic potential at r is related to the electron density of a neutral atom by the density functional

where c = 4.7854. This expression has been used to derive an approximate value for the total energy of an atom in terms of the potential at the nuclear position

in which k is 3/7 = 0.4286, compared with the exact value for the hydrogen atom of 0.5. In an extension of this expression to molecules, due to Politzer (1979), the total electronic energy of a molecule is expressed as

in which tne effective Born coefficient, defined by U = -NAn'"/a(l - l/n eff ); and the lattice energy U. The discrepancy between the x-refinement results and the experimental value for U is 0.92%. The exact value of the lattice energy is very sensitive to the amount of charge transfer. If the charge transfer is reduced by only one standard deviation to 0.94 e, the agreement with the experimental lattice energy is within 0.12%. Conversely, an increase by one standard deviation worsens the discrepancy. Nevertheless, the agreement with the calorimetric lattice energy is satisfying, given the simplicity of the treatment of the nonelectrostatic interactions.


203

9.4 The Cohesive Energy of Molecular Crystals 9.4.1 Molecular Interactions

Short-range repulsive forces are a direct result of the Pauli exclusion principle and are thus quantum mechanical in nature. Kitaigorodskii (1961) has emphasized that such short-range repulsive forces play a major role in determining the packing in molecular crystals. The size and shape of molecules is determined by the repulsive forces, and the molecules pack as closely as is permitted by these forces. Repulsive forces determine, for example, the melting point of a solid. Whenever the packing is efficient, the melting point tends to be high. The attractive forces, on the other hand, govern the heat of vaporization and therefore the boiling point. Trouton's rule, which relates the normal boiling point of a liquid to its heat of vaporization, is a manifestation of this relation. Like the Coulombic forces, the van der Waals interactions decrease less rapidly with increasing distance than the repulsive forces. They include interactions that arise from the dipole moments induced by nearby charges and permanent dipoles, as well as interactions between instantaneous dipole moments, referred to as dispersion forces (Israelachvili 1992). Instantaneous dipole moments can be thought of as arising from the motions of the electrons. Even though the electron probability distribution of a spherical atom has its center of gravity at the nuclear position, at any very short instance the electron positions will generally not be centered on the nucleus. Quantitative treatment of the interaction between two identical Bohr atoms, consisting of point electrons and nuclei, leads to an expression which, apart from a numerical factor, is the same as that derived quantum-mechanically by London (1937):

Here,xo0 an and I are tne polarizability and the ionization energy of the atom,

respectively. The r6 dependence is also encountered in the so-called Debye interaction between a permanent dipole and an induced dipole, given by

r

The dipole-dipole interactions, frequently referred to as Keesom interactions, are historically included in the van der Waals interactions, even though they are purely electrostatic. For molecules that are free to orient themselves, the dipoledipole interactions must be averaged over the molecular orientations, as the angular dependence of the interaction energy is comparable to the Boltzmann energy kBT (Israelachvili 1992, p. 62). With the averaging of the Keesom

204


interactions, the total van der Waals energy for two dissimilar bodies is given by

When the interaction is expressed in terms of pairwise atom-atom potential functions, all three components of the van der Waals interactions are grouped together because of their common r~6 distance dependence. A repulsive term is added, while Coulombic interactions may be accounted for separately. In the expression due to Lennard Jones, the repulsion has an r ~ 1 2 dependence, to give the pairwise potential function

in which A and B are specific for the type of atoms. Alternatively, the repulsion is frequently described by an exponential term, or

in which A, B, and C are, again, element specific. For interactions between unlike atoms i and j, the coefficients A, B, and C can be derived from the coefficients of the homoatomic interactions by approximately valid combining rules, defined as

The interaction energy between two molecules or molecular fragments is obtained as a sum over all pairwise atom-atom interactions. The atom-atom potential expressions implicitly assume that the interactions are two-body interactions, undisturbed by other bodies in the vicinity, and that they are isotropic about the atomic centers. The pairwise interaction may be obtained using density functional theory. Let pA and pB be the density for individual A and B subsystems. In terms of density functional theory, the interaction energy EAB consists of correlation, kinetic energy, and exchange and electrostatic contributions, or


205

FIG. 9.2 Nonbonded potential for O—O. Broken line: according to the Gordon-Kim density functional model augmented with an attractive potential (Spackman 1986). Full line: empirical curve based on a fit to oxohydrocarbon crystal structures (Cox et al. 1981).

in which A£electrostatic represents the Coulombic interactions, and the ThomasFermi-Dirac model [Eqs. (9.7) and (9.8)] has been used.4 This is the Gordon-Kim (1972) model. A number of applications have demonstrated that the Gordon-Kim model leads to a qualitatively valid description of potential energy surfaces between closed-shell subsystems. Spackman (1986a) has used the Gordon-Kim model, with an empirical scaling parameter for the exchange contribution, to derive a set of short-range repulsive potentials for homoatomic pairs up to the element krypton. The resulting potentials are complemented with an attractive r~6 term representing the dispersion forces. The Gordon-Kim interaction functions may be compared with empirical potential functions derived by energy- or net-force minimization methods using known crystal structures. The O—O Gordon-Kim potentials are more repulsive, as illustrated in Fig. 9.2. Spackman points out that the empirical potentials likely contain a significant attractive component because of the inadequate allowance for electrostatic interactions in their derivation. This attractive component is included in the electrostatic interaction in the density functional model. The application of the Gordon-Kim model to open-shell systems, which must include the interaction between unpaired spins, appears less successful (Kim and Gordon 1974). Expressions of the forms in Eqs. (9.45) and (9.46) have been extremely useful in molecular modeling studies. An extensive literature exists on the choice of

4

Note that in this equation an extra term is included to account for correlation energy. The term is derived from the topology of p.

206


coefficients and on the application of the expressions in structure calculations (Wampler 1994, Cornell et al. 1995). 9.4.2 Interactions Between Molecules in Crystals Like the interaction energy between two molecules, the total lattice energy of a molecular crystal contains several contributions from the different types of interactions. We may write

In this expression, the dipole-dipole interactions are included in the electrostatic term rather than in the van der Waals interactions as in Eq. (9.43). Of the four contributions, the electrostatic energy can be derived directly from the charge distribution. As discussed in section 9.2, information on the nonelectrostatic terms can be deduced indirectly from the charge density. The polarizability a, which occurs in the expressions for the Debye and dispersion terms of Eqs. (9.41) and (9.42), can be expressed as a functional of the density (Matsuzawa and Dixon 1994), and also obtained from the quadrupole moments of the experimental charge density distribution (see section 12.3.2). However, most frequently, empirical atom-atom pair potential functions like Eqs. (9.45) and (9.46) are used in the calculation of the nonelectrostatic contributions to the intermolecular interactions. 9.4.2.1 Hydrogen-bonding Hydrogen-bonding is one of the prime interactions determining the packing motif in molecular crystals. As discussed in section 12.3.3, topological analysis of the total charge distribution indicates normal hydrogen-bonding to be a closed-shell interaction, with a very low density at the critical point, and V2p at the critical point being invariably positive. But for very short hydrogen-bonds, covalent contributions will be of increasing importance. Nevertheless, simple electrostatic models can to a large extent explain the energetics of hydrogen-bond formation. In the work of Spackman (1986b), hydrogen-bonding is accounted for by electrostatic forces, combined with the omission of the repulsive term in the hydrogen-acceptor interaction. 9.4.3 Expressions for the Evaluation of the Electrostatic Contribution to the Lattice Energy of Molecular Crystals The electrostatic energy of a molecular crystal can be evaluated with summation over the structure factors in Eq. (9.15). But to obtain the cohesive energy of a molecular crystal with such a summation, we would have to subtract the molecular electrostatic energies, which are implicitly included in the result. An alternative is to perform the calculation in direct space. With the molecular densities pA and pB obtained with one of the partitioning


207

methods discussed in chapter 6, the electrostatic interaction energy is given by

where A stands for a unique central molecule, and B represents the molecules in the remainder of the crystal. If there is more than one molecule in the asymmetric unit, a term for the interaction of each additional molecule, A', with its environment, is to be added. The doubly counted A-A' interactions must be subtracted. Expression (9.49) can be evaluated by substituting the expression for the potential in terms of the charge distribution [Eq. (8.2)], which gives

For a slowly varying charge distribution, the potential can be expanded in a Taylor series with $(0), the potential at the origin of the distribution B, as leading term (Buckingham 1959, 1970, 1978; Jackson 1974):

Since, following Eq. (8.4), — V3d and 0|igand orbitals in the LCAO expression. If the ligand orbital is much lower in energy, the bonding will be weaker, and the a-donation into the metal atomic orbital will be less important. If the transition metal atom has more than the six d electrons indicated on the diagram, the antibonding 2eg molecular orbitals will also be populated and the metal-ligand bonding will be weakened. An example of this quite-prevalent effect is encountered in the series FeS2, CoS2, NiS2, discussed in section 10.4.2. The crystal-field-stabilized metal orbitals can combine with empty n orbitals on suitable ligands, leading to rc-bonding, which is not included in Fig. 10.4. Since the ligand orbitals participating in this molecular orbital are generally empty, electrons are transferred to the ligands by so-called Tt-back-donation. This effect will lead to a lower population than predicted electrostatically for the crystal-fieldstabilized metal orbitals, and, if the donation is into a ligand antibonding orbital, it leads to a lengthening of the bond on the ligand. 10.2.3 The Relation Between the Occupancies of Transition-Metal Valence Orbitals and the Multipole Population Parameters In general, the atom-centered density model functions describe both the one-center valence density and the two-center density resulting from overlap of the valence orbitals. In the case of transition metal complexes, the overlap density in the metal-ligand bond is frequently quite small. An ab-initio theoretical calculation of the cobalt-porphine complex with a Co-N distance of 1.987 A, for instance, gives, for the 2Alg ground-state, a- and Tt-metal-ligand overlap populations of 0.02 and 0.04 e, respectively (Kashiwagi et al. 1978). This means that we can equate, to a good approximation, the multipolar density centered on the transition metal atom with the population of the outer valence shells of the atom. If (t)(d,) is the atomic d-orbital basis set, the 3d density can be written as

The cross terms (t)(d,)<j)(dj), with i = j in Eq. (10.3) do not appear in the case of the isolated atom for which the electron density is the sum of the square of the atomic orbitals. In the molecular case, the cross terms will only be nonzero for orbitals belonging to the same representation of the point group of the molecule, like the eg orbitals in the case of trigonal site symmetry discussed above. In the square-planar point group D4h(4/m mm), the orbitals have a l g , blg, b2g, and eg symmetry, and no such mixing occurs. We recall that in the multipolar expansion, the 3d density is expressed in terms of the density-normalized spherical harmonic functions dlmp as

Since the d orbitals are invariant with respect to inversion through the nuclear position, only / even terms will occur in this summation.

Charge Density Studies of Transition Metal Compounds

217

Equating Eqs. (10.3) and (10.4) leads to an especially simple expression if the radial dependence of the density is equal in both descriptions. In this case, the radial parts cancel and we obtain Here, Y(j- is the 15-element column vector of the angular part of the 4>(dt)(j)(dj) orbital products, Py is the row vector of the 15 unique elements of the symmetric 5x5 matrix of the coefficients in Eq. (10.3), and Plmp is the row vector containing the coefficients of the 15 spherical harmonic density functions d,mp with / = 0, 2, or 4. Density functions with other / values do not contribute to the d-orbital density. The spherical harmonic functions constitute a complete set of functions in the spherical point group. A product of two spherical harmonics such as ytyj must therefore be a linear combination of spherical harmonic functions. An example of such an expression is For our purpose, it is preferable to express the right-hand side of this equation in terms of the spherical harmonic density functions. Use of the ratio of orbital- and density-function normalization factors gives the result Expression (10.6) shows that the product of two identical dZ2 orbitals contains hexadecapolar, quadrupolar, and monopolar density functions. The product equations can, in general, be written as Y;J. = Ld,mp. The elements of the matrix L are the coefficients in expressions like Eq. (10.6). A complete set of the equations for / < 2 is given in Table E.3 of appendix E. The equivalence for the density in Eq. (10.5) can then be written as We obtain, for the relation between the coefficients P^ and P,mp, or

The d-orbital occupancies are derived from the experimental multipole populations by the inverse expression (Holladay et al. 1983) The matrix M-1 is given in appendix I. In all but triclinic point groups, site-symmetry restrictions limit the allowed functions beyond the / even requirement. The symmetry-allowed multipolar density functions are given by the "index-picking" rules of appendix D, section D.3, and are listed in Table 10.1. The M""1 matrices specific for higher point groups are obtained by omission of symmetry-forbidden columns in the full 1 5 x 1 5 matrix. This leads to rows with zero elements for the nonallowed cross products between d orbitals, which are subsequently omitted to recover a reduced matrix. The matrix for the point group DM = 4/m mm is shown as an example in Table 10.2.

218


T A B L E 10.1 Symmetry-Allowed Multipole Functions Describing d-Orbital Density Allowed Values of /, m + of M

Poinl Group

l,T

'a

Dimension of M

/ = 0, 2, 4, all m 00, 20. 22 + , 22-, 40, 42 + , 42-, 44 + , 4400,20, 22 + , 40, 42 + , 44 + 00, 20, 40, 44 + ,4400, 20, 40, 44 + 00, 20,40,43 + , 4300, 20, 40, 43 + 00, 20, 40 00, 0.78245 (40 + ) + 0.57939 (44 + )c

2, HI, 2/m

222, m2m, mmm 4, 4/m, 4 422, 42m, 4mm, 4/mmm

3,3

32, 3m, 3m 6, 6, 6/m, 622, 6mm, 6 m2, 6/mmm 23, m3, 432, 43m, m3m

15 x 15 9x9 6x6 5 x 5 (4 x

4x4

5 x 5 (4 x 4x4 3x3 2x2

4) b 4) b

" Principle symmetry axis is z axis. b Dimension can be reduced by rotation of coordinate system; see text. c This function is usually described as the cubic harmonic K4 (see appendix D).

10.2.3.1 The Relationships in Terms of Symmetry-Adapted Orbitals For the point group D4h, the atomic d-orbital functions belong to four different group-theoretical representations. When the same representation occurs more than once, as it does, for example, in trigonal point groups, M"1 will contain cross terms between orbitals of the same symmetry, as shown in Table 10.3(a). In this case, we are interested in the population of the symmetry-adapted orbitals ysk, such as defined for the trigonal case by expression (10.2). The symmetry-adapted orbitals are linear combinations of the original functions, that is,

while the density products are given by

The populations Pskl of the symmetry-adapted orbital products follow, in analogy to Eq. (10.7), from

T A B L E 10.2 The Matrix IV1 ', Defininj; Orbital-Miultipole Relati ons, for the Pioint

Group D 4h Descriiption z2

P20 f*2 1 -f > P22 +

P22-

"it-

xz, yz x2 - y2

xy

Source: Coppens and Becker (1992:).

"„

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